The document discusses exponential and logarithmic functions. It defines exponential functions as functions of the form f(x) = bx, where b is the base. It provides examples of graphs of exponential functions with different bases. It then introduces logarithmic functions as the inverse of exponential functions, where logarithms are defined as logbx = y if x = by. It provides properties and examples involving logarithmic functions.
Module 4 exponential and logarithmic functionsdionesioable
This document provides an overview of a module on logarithmic functions. It discusses the definition of logarithmic functions as the inverse of exponential functions, how to graph logarithmic functions by reflecting the graph of the corresponding exponential function across the line y=x, and properties of logarithmic function graphs like their domains, ranges, asymptotes, and behavior. It also covers laws of logarithms and how to solve logarithmic equations. The document is designed to teach students to define logarithmic functions, graph them, use laws of logarithms, and solve simple logarithmic equations.
logarithmic, exponential, trigonometric functions and their graphs.pptYohannesAndualem1
Introduction:
[Start with a brief introduction about yourself, including your profession or main area of expertise.]
Background:
[Discuss your background, education, and any relevant experiences that have shaped your journey.]
Accomplishments:
[Highlight notable achievements, awards, or significant projects you've been involved in.]
Expertise:
[Detail your areas of expertise, skills, or specific knowledge that sets you apart in your field.]
Passions and Interests:
[Share your passions, hobbies, or interests outside of your professional life, adding depth to your personality.]
Vision or Mission:
[If applicable, articulate your vision, mission, or goals in your chosen field or in life in general.]
Closing Statement:
[End with a closing statement that summarizes your essence or leaves a lasting impression.]
Feel free to customize each section with your own personal details and experiences. If you need further assistance or have specific points you'd like to include, feel free to let me know!
This document discusses derivatives of various functions including:
- Exponential functions like ex and ax where the derivative of ex is ex and the derivative of ax is axln(a)
- Inverse functions where the derivative of the inverse is the reciprocal of the derivative of the original function
- Logarithmic functions like ln(x), loga(x) where the derivatives are 1/x and 1/(xln(a))
- Using logarithmic differentiation to find derivatives of functions like f(x)g(x)
It also provides practice problems finding derivatives of various functions and solving related equations.
1. The document discusses exponential and logarithmic functions, including: defining them, understanding their inverse relationship, and how they can model real-world situations like population growth.
2. It provides examples of exponential growth and decay functions, defines the important constant e, and covers the basic properties and graphs of exponential and logarithmic functions.
3. The document also discusses taking derivatives of exponential and logarithmic functions, using rules like the power rule, exponential rule, and logarithmic rule.
دالة الاكسبونيشل الرياضية والفريدة من نوعها و كيفية استخدامهالzeeko4
The document discusses derivatives of exponential and logarithmic functions. It begins by introducing exponential functions and their role in modeling growth and decay. It then derives the derivative of the exponential function ex as ex and the derivative of the natural logarithm function ln x as 1/x. Finally, it generalizes these results to derivatives of exponential and logarithmic functions with arbitrary bases b > 0 and b ≠ 1.
This document discusses exponential and logarithmic functions. It defines exponential functions as functions of the form f(x) = bx where b is the base, and defines logarithmic functions as the inverses of exponential functions. Properties of exponential and logarithmic functions are presented, including their domains, ranges, and asymptotes. Examples of graphing common exponential and logarithmic functions are shown. Methods for solving exponential and logarithmic equations are also provided.
This document discusses several topics in calculus of several variables:
- Functions of several variables and their partial derivatives
- Maxima and minima of functions of several variables
- Double integrals and constrained maxima/minima using Lagrange multipliers
It provides examples of computing partial derivatives of functions, interpreting them geometrically, and using partial derivatives to determine rates of change. Level curves are also discussed as a way to sketch graphs of functions of two variables.
The document defines logarithmic functions and provides examples of converting between logarithmic and exponential forms. It discusses the properties of logarithmic equality and solving logarithmic equations by rewriting them in exponential form. Key points include: logarithmic and exponential forms are inverses; the exponent becomes the logarithm; logarithmic functions have a domain of positive real numbers; and solving logarithms involves setting arguments equal when bases are the same.
Module 4 exponential and logarithmic functionsdionesioable
This document provides an overview of a module on logarithmic functions. It discusses the definition of logarithmic functions as the inverse of exponential functions, how to graph logarithmic functions by reflecting the graph of the corresponding exponential function across the line y=x, and properties of logarithmic function graphs like their domains, ranges, asymptotes, and behavior. It also covers laws of logarithms and how to solve logarithmic equations. The document is designed to teach students to define logarithmic functions, graph them, use laws of logarithms, and solve simple logarithmic equations.
logarithmic, exponential, trigonometric functions and their graphs.pptYohannesAndualem1
Introduction:
[Start with a brief introduction about yourself, including your profession or main area of expertise.]
Background:
[Discuss your background, education, and any relevant experiences that have shaped your journey.]
Accomplishments:
[Highlight notable achievements, awards, or significant projects you've been involved in.]
Expertise:
[Detail your areas of expertise, skills, or specific knowledge that sets you apart in your field.]
Passions and Interests:
[Share your passions, hobbies, or interests outside of your professional life, adding depth to your personality.]
Vision or Mission:
[If applicable, articulate your vision, mission, or goals in your chosen field or in life in general.]
Closing Statement:
[End with a closing statement that summarizes your essence or leaves a lasting impression.]
Feel free to customize each section with your own personal details and experiences. If you need further assistance or have specific points you'd like to include, feel free to let me know!
This document discusses derivatives of various functions including:
- Exponential functions like ex and ax where the derivative of ex is ex and the derivative of ax is axln(a)
- Inverse functions where the derivative of the inverse is the reciprocal of the derivative of the original function
- Logarithmic functions like ln(x), loga(x) where the derivatives are 1/x and 1/(xln(a))
- Using logarithmic differentiation to find derivatives of functions like f(x)g(x)
It also provides practice problems finding derivatives of various functions and solving related equations.
1. The document discusses exponential and logarithmic functions, including: defining them, understanding their inverse relationship, and how they can model real-world situations like population growth.
2. It provides examples of exponential growth and decay functions, defines the important constant e, and covers the basic properties and graphs of exponential and logarithmic functions.
3. The document also discusses taking derivatives of exponential and logarithmic functions, using rules like the power rule, exponential rule, and logarithmic rule.
دالة الاكسبونيشل الرياضية والفريدة من نوعها و كيفية استخدامهالzeeko4
The document discusses derivatives of exponential and logarithmic functions. It begins by introducing exponential functions and their role in modeling growth and decay. It then derives the derivative of the exponential function ex as ex and the derivative of the natural logarithm function ln x as 1/x. Finally, it generalizes these results to derivatives of exponential and logarithmic functions with arbitrary bases b > 0 and b ≠ 1.
This document discusses exponential and logarithmic functions. It defines exponential functions as functions of the form f(x) = bx where b is the base, and defines logarithmic functions as the inverses of exponential functions. Properties of exponential and logarithmic functions are presented, including their domains, ranges, and asymptotes. Examples of graphing common exponential and logarithmic functions are shown. Methods for solving exponential and logarithmic equations are also provided.
This document discusses several topics in calculus of several variables:
- Functions of several variables and their partial derivatives
- Maxima and minima of functions of several variables
- Double integrals and constrained maxima/minima using Lagrange multipliers
It provides examples of computing partial derivatives of functions, interpreting them geometrically, and using partial derivatives to determine rates of change. Level curves are also discussed as a way to sketch graphs of functions of two variables.
The document defines logarithmic functions and provides examples of converting between logarithmic and exponential forms. It discusses the properties of logarithmic equality and solving logarithmic equations by rewriting them in exponential form. Key points include: logarithmic and exponential forms are inverses; the exponent becomes the logarithm; logarithmic functions have a domain of positive real numbers; and solving logarithms involves setting arguments equal when bases are the same.
This document introduces exponents, logarithms, and exponential and logarithmic equations. It defines exponents and logarithms, including common, natural, and base conversion. Laws of exponents and logarithms are presented along with properties of radicals, rational exponents, negative exponents, and zero exponents. Examples of solving exponential and logarithmic equations are provided. The summary provides an overview of the key topics and concepts covered in the document.
This is meant for university students taking either information technology or engineering courses, this course of differentiation, Integration and limits helps you to develop your problem solving skills and other benefits that come along with it.
The document discusses functions and relations. It defines functions, relations, and domain and range. It provides examples of expressing relations in set notation, tabular form, equations, graphs, and mappings. It also discusses evaluating, adding, multiplying, dividing, and composing functions. Graphs of various functions like absolute value, piecewise, greatest integer, and least integer functions are also explained.
1) The document discusses solving exponential and logarithmic equations using logarithms and exponentiation. It provides examples of evaluating logarithmic functions, using properties of logarithms, and changing logarithmic bases.
2) An example solves the equation for when health costs in the US will reach $250 billion according to an exponential model for annual costs. The answer is about 21.66 years after 1960.
3) Another example solves an exponential equation to find the year when populations of India and China will be equal according to exponential growth models, finding it will be around 2028.
1. Logarithmic functions are the inverse of exponential functions. The logarithm of a number represents the power to which the base must be raised to produce that number.
2. Common properties of logarithms include adding exponents when multiplying terms with the same base and subtracting exponents when dividing terms with the same base.
3. The change of base formula allows converting between logarithms with different bases using a ratio of logarithms and the natural logarithm.
This document provides an overview of exponential and logarithmic functions. It defines one-to-one functions and inverse functions. It explains how to find the inverse of a one-to-one function and shows that the inverse of f(x) is f-1(x). Properties of exponential functions like f(x)=ax and logarithmic functions like f(x)=logax are described. The product, quotient, and power rules for logarithms are outlined along with examples. Finally, it discusses how to solve exponential and logarithmic equations using properties of these functions.
This document discusses exponential functions. It defines exponential functions as functions where f(x) = ax + B, where a is a real constant and B is any expression. Examples of exponential functions given are f(x) = e-x - 1 and f(x) = 2x. The document also discusses evaluating exponential equations with like and different bases using logarithms. It provides examples of graphing exponential functions and discusses the key characteristics of functions of the form f(x) = bx, including their domains, ranges, and asymptotic behavior. The document concludes with an example of applying exponential functions to model compound interest.
This document defines functions and discusses key concepts related to functions including:
- A function relates each element of its domain to a unique element of its range.
- Functions can be one-to-one or many-to-one.
- Functions are represented in set notation, tabular form, equations, and graphs.
- The domain of a function is the set of possible inputs, and the range is the set of possible outputs.
The document discusses properties of logarithms. It begins by recalling rules of exponents and their corresponding rules of logarithms. Four basic logarithm rules are presented: 1) logb(1) = 0, 2) logb(xy) = logb(x) + logb(y), 3) logb(x/y) = logb(x) - logb(y), 4) logb(xt) = tlogb(x). It then works through an example problem to demonstrate using these rules to write the logarithm of a expression in terms of logarithms of its variables. It concludes by noting that logarithms and exponentials are inverse functions, so logb(bx) =
This document discusses the natural logarithm (ln) and its relationship to the mathematical constant e. It provides examples of how to calculate ln of different values with and without a calculator. It also describes laws of logarithms like ln(ab) = ln(a) + ln(b) and how to solve exponential equations using logarithms by changing the operation to multiplication or division. Examples are given of finding the value of x in equations involving natural logs and the constant e.
This document covers exponential and logarithmic functions, including their definitions, graphs, properties, and applications. Some key points are:
- Exponential functions take the form f(x) = bx where b is the base. Their graphs stretch upwards and have asymptotes at the x-axis.
- The natural exponential function is f(x) = ex.
- Logarithmic functions are the inverses of exponential functions and take the form f(x) = logbx. Their graphs stretch upwards and have asymptotes at the y-axis.
- Properties of exponents and logarithms include the change of base formula.
- Exponential and logarithmic equations can be solved by isolating the exponential/log
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
This document provides an overview of exponential and logarithmic functions. It covers composite and inverse functions, exponential functions, logarithmic functions, properties of logarithms, common logarithms, exponential and logarithmic equations, and natural exponential and logarithmic functions. Example problems are provided to illustrate each concept.
This document covers exponential and logarithmic functions, including their definitions, graphs, properties, and applications. Some key points are:
- Exponential functions have the form f(x) = bx where b is the base. Their graphs stretch upwards and have asymptotes at the x-axis.
- The natural exponential function is f(x) = ex.
- Logarithmic functions are the inverses of exponential functions and have the form f(x) = logbx. Their graphs stretch downwards and have asymptotes at the y-axis.
- Properties of exponents and logarithms include the change of base formula and rules for simplifying expressions.
- Exponential and logarithmic equations can be solved
This document discusses solving numerical equations involving logarithmic and exponential functions. It provides examples of solving both log equations and exponential equations. To solve log equations, the problems are rewritten in exponential form by removing the log. To solve exponential equations, the exponents are brought down by rewriting the problem in logarithmic form. Steps include isolating the exponential or logarithmic term containing the unknown, rewriting the equation accordingly, and then solving for the unknown variable. Practice problems with solutions are provided to illustrate the process.
The document discusses functions and evaluating functions. It provides examples of determining if a given equation is a function using the vertical line test and evaluating functions by substituting values into the function equation. It also includes examples of evaluating composite functions using flow diagrams to illustrate the steps of evaluating each individual function.
This document provides a summary of precalculus concepts including:
1. Functions and their graphs including function definitions, transformations, combinations, and compositions of functions.
2. Trigonometry including trigonometric functions, graphs of trigonometric functions, and trigonometric identities.
3. Graphs of second-degree equations including circles, parabolas, ellipses, and hyperbolas.
The document contains examples and explanations of key precalculus topics to serve as a review for a Math 131 course. It covers essential functions like polynomials, rational functions, and transcendental functions. It also discusses trigonometric functions and their graphs along with transformations of functions.
This document discusses solving numerical equations involving logarithmic and exponential functions. It provides examples of solving:
1) Logarithmic equations by dropping the log and writing the equation in exponential form.
2) Exponential equations by isolating the exponential term containing the unknown, then taking the log of both sides to write it in logarithmic form.
3) The document demonstrates solving sample equations of each type step-by-step and explains the differences between logarithmic and exponential equations.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
This document introduces exponents, logarithms, and exponential and logarithmic equations. It defines exponents and logarithms, including common, natural, and base conversion. Laws of exponents and logarithms are presented along with properties of radicals, rational exponents, negative exponents, and zero exponents. Examples of solving exponential and logarithmic equations are provided. The summary provides an overview of the key topics and concepts covered in the document.
This is meant for university students taking either information technology or engineering courses, this course of differentiation, Integration and limits helps you to develop your problem solving skills and other benefits that come along with it.
The document discusses functions and relations. It defines functions, relations, and domain and range. It provides examples of expressing relations in set notation, tabular form, equations, graphs, and mappings. It also discusses evaluating, adding, multiplying, dividing, and composing functions. Graphs of various functions like absolute value, piecewise, greatest integer, and least integer functions are also explained.
1) The document discusses solving exponential and logarithmic equations using logarithms and exponentiation. It provides examples of evaluating logarithmic functions, using properties of logarithms, and changing logarithmic bases.
2) An example solves the equation for when health costs in the US will reach $250 billion according to an exponential model for annual costs. The answer is about 21.66 years after 1960.
3) Another example solves an exponential equation to find the year when populations of India and China will be equal according to exponential growth models, finding it will be around 2028.
1. Logarithmic functions are the inverse of exponential functions. The logarithm of a number represents the power to which the base must be raised to produce that number.
2. Common properties of logarithms include adding exponents when multiplying terms with the same base and subtracting exponents when dividing terms with the same base.
3. The change of base formula allows converting between logarithms with different bases using a ratio of logarithms and the natural logarithm.
This document provides an overview of exponential and logarithmic functions. It defines one-to-one functions and inverse functions. It explains how to find the inverse of a one-to-one function and shows that the inverse of f(x) is f-1(x). Properties of exponential functions like f(x)=ax and logarithmic functions like f(x)=logax are described. The product, quotient, and power rules for logarithms are outlined along with examples. Finally, it discusses how to solve exponential and logarithmic equations using properties of these functions.
This document discusses exponential functions. It defines exponential functions as functions where f(x) = ax + B, where a is a real constant and B is any expression. Examples of exponential functions given are f(x) = e-x - 1 and f(x) = 2x. The document also discusses evaluating exponential equations with like and different bases using logarithms. It provides examples of graphing exponential functions and discusses the key characteristics of functions of the form f(x) = bx, including their domains, ranges, and asymptotic behavior. The document concludes with an example of applying exponential functions to model compound interest.
This document defines functions and discusses key concepts related to functions including:
- A function relates each element of its domain to a unique element of its range.
- Functions can be one-to-one or many-to-one.
- Functions are represented in set notation, tabular form, equations, and graphs.
- The domain of a function is the set of possible inputs, and the range is the set of possible outputs.
The document discusses properties of logarithms. It begins by recalling rules of exponents and their corresponding rules of logarithms. Four basic logarithm rules are presented: 1) logb(1) = 0, 2) logb(xy) = logb(x) + logb(y), 3) logb(x/y) = logb(x) - logb(y), 4) logb(xt) = tlogb(x). It then works through an example problem to demonstrate using these rules to write the logarithm of a expression in terms of logarithms of its variables. It concludes by noting that logarithms and exponentials are inverse functions, so logb(bx) =
This document discusses the natural logarithm (ln) and its relationship to the mathematical constant e. It provides examples of how to calculate ln of different values with and without a calculator. It also describes laws of logarithms like ln(ab) = ln(a) + ln(b) and how to solve exponential equations using logarithms by changing the operation to multiplication or division. Examples are given of finding the value of x in equations involving natural logs and the constant e.
This document covers exponential and logarithmic functions, including their definitions, graphs, properties, and applications. Some key points are:
- Exponential functions take the form f(x) = bx where b is the base. Their graphs stretch upwards and have asymptotes at the x-axis.
- The natural exponential function is f(x) = ex.
- Logarithmic functions are the inverses of exponential functions and take the form f(x) = logbx. Their graphs stretch upwards and have asymptotes at the y-axis.
- Properties of exponents and logarithms include the change of base formula.
- Exponential and logarithmic equations can be solved by isolating the exponential/log
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
This document provides an overview of exponential and logarithmic functions. It covers composite and inverse functions, exponential functions, logarithmic functions, properties of logarithms, common logarithms, exponential and logarithmic equations, and natural exponential and logarithmic functions. Example problems are provided to illustrate each concept.
This document covers exponential and logarithmic functions, including their definitions, graphs, properties, and applications. Some key points are:
- Exponential functions have the form f(x) = bx where b is the base. Their graphs stretch upwards and have asymptotes at the x-axis.
- The natural exponential function is f(x) = ex.
- Logarithmic functions are the inverses of exponential functions and have the form f(x) = logbx. Their graphs stretch downwards and have asymptotes at the y-axis.
- Properties of exponents and logarithms include the change of base formula and rules for simplifying expressions.
- Exponential and logarithmic equations can be solved
This document discusses solving numerical equations involving logarithmic and exponential functions. It provides examples of solving both log equations and exponential equations. To solve log equations, the problems are rewritten in exponential form by removing the log. To solve exponential equations, the exponents are brought down by rewriting the problem in logarithmic form. Steps include isolating the exponential or logarithmic term containing the unknown, rewriting the equation accordingly, and then solving for the unknown variable. Practice problems with solutions are provided to illustrate the process.
The document discusses functions and evaluating functions. It provides examples of determining if a given equation is a function using the vertical line test and evaluating functions by substituting values into the function equation. It also includes examples of evaluating composite functions using flow diagrams to illustrate the steps of evaluating each individual function.
This document provides a summary of precalculus concepts including:
1. Functions and their graphs including function definitions, transformations, combinations, and compositions of functions.
2. Trigonometry including trigonometric functions, graphs of trigonometric functions, and trigonometric identities.
3. Graphs of second-degree equations including circles, parabolas, ellipses, and hyperbolas.
The document contains examples and explanations of key precalculus topics to serve as a review for a Math 131 course. It covers essential functions like polynomials, rational functions, and transcendental functions. It also discusses trigonometric functions and their graphs along with transformations of functions.
This document discusses solving numerical equations involving logarithmic and exponential functions. It provides examples of solving:
1) Logarithmic equations by dropping the log and writing the equation in exponential form.
2) Exponential equations by isolating the exponential term containing the unknown, then taking the log of both sides to write it in logarithmic form.
3) The document demonstrates solving sample equations of each type step-by-step and explains the differences between logarithmic and exponential equations.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Physiology and chemistry of skin and pigmentation, hairs, scalp, lips and nail, Cleansing cream, Lotions, Face powders, Face packs, Lipsticks, Bath products, soaps and baby product,
Preparation and standardization of the following : Tonic, Bleaches, Dentifrices and Mouth washes & Tooth Pastes, Cosmetics for Nails.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
-------------------------------------------------------------------------------
Find out more about ISO training and certification services
Training: ISO/IEC 27001 Information Security Management System - EN | PECB
ISO/IEC 42001 Artificial Intelligence Management System - EN | PECB
General Data Protection Regulation (GDPR) - Training Courses - EN | PECB
Webinars: https://pecb.com/webinars
Article: https://pecb.com/article
-------------------------------------------------------------------------------
For more information about PECB:
Website: https://pecb.com/
LinkedIn: https://www.linkedin.com/company/pecb/
Facebook: https://www.facebook.com/PECBInternational/
Slideshare: http://www.slideshare.net/PECBCERTIFICATION
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
हिंदी वर्णमाला पीपीटी, hindi alphabet PPT presentation, hindi varnamala PPT, Hindi Varnamala pdf, हिंदी स्वर, हिंदी व्यंजन, sikhiye hindi varnmala, dr. mulla adam ali, hindi language and literature, hindi alphabet with drawing, hindi alphabet pdf, hindi varnamala for childrens, hindi language, hindi varnamala practice for kids, https://www.drmullaadamali.com
Thinking of getting a dog? Be aware that breeds like Pit Bulls, Rottweilers, and German Shepherds can be loyal and dangerous. Proper training and socialization are crucial to preventing aggressive behaviors. Ensure safety by understanding their needs and always supervising interactions. Stay safe, and enjoy your furry friends!
3. Exponential Function
The function defined by
is called an exponential function with base b
and exponent x.
The domain of f is the set of all real numbers.
( ) ( 0, 1)
x
f x b b b
4. Example
The exponential function with base 2 is the function
with domain (– , ).
The values of f(x) for selected values of x follow:
( ) 2x
f x
(3)
f
3
2
f
(0)
f
3
2 8
3/2 1/2
2 2 2 2 2
0
2 1
5. Example
The exponential function with base 2 is the function
with domain (– , ).
The values of f(x) for selected values of x follow:
( ) 2x
f x
( 1)
f
2
3
f
1 1
2
2
2/3
2/3 3
1 1
2
2 4
6. Laws of Exponents
Let a and b be positive numbers and let x
and y be real numbers. Then,
1.
2.
3.
4.
5.
x y x y
b b b
x
x y
y
b
b
b
y
x xy
b b
x x x
ab a b
x x
x
a a
b b
7. Examples
Let f(x) = 2
2x – 1
. Find the value of x for which f(x) = 16.
Solution
We want to solve the equation
2
2x – 1
= 16 = 2
4
But this equation holds if and only if
2x – 1 = 4
giving x = .
5
2
8. Examples
Sketch the graph of the exponential function f(x) = 2x.
Solution
First, recall that the domain of this function is the set of
real numbers.
Next, putting x = 0 gives y = 20 = 1, which is the y-intercept.
(There is no x-intercept, since there is no value of x for
which y = 0)
9. Examples
Sketch the graph of the exponential function f(x) = 2x.
Solution
Now, consider a few values for x:
Note that 2x approaches zero as x decreases without bound:
✦ There is a horizontal asymptote at y = 0.
Furthermore, 2x increases without bound when x increases
without bound.
Thus, the range of f is the interval (0, ).
x – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5
y 1/32 1/16 1/8 1/4 1/2 1 2 4 8 16 32
10. Examples
Sketch the graph of the exponential function f(x) = 2x.
Solution
Finally, sketch the graph:
x
y
– 2 2
4
2
f(x) = 2x
11. Examples
Sketch the graph of the exponential function f(x) = (1/2)x.
Solution
First, recall again that the domain of this function is the
set of real numbers.
Next, putting x = 0 gives y = (1/2)0 = 1, which is the
y-intercept.
(There is no x-intercept, since there is no value of x for
which y = 0)
12. Examples
Sketch the graph of the exponential function f(x) = (1/2)x.
Solution
Now, consider a few values for x:
Note that (1/2)x increases without bound when x decreases
without bound.
Furthermore, (1/2)x approaches zero as x increases without
bound: there is a horizontal asymptote at y = 0.
As before, the range of f is the interval (0, ).
x – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5
y 32 16 8 4 2 1 1/2 1/4 1/8 1/16 1/32
13. Examples
Sketch the graph of the exponential function f(x) = (1/2)x.
Solution
Finally, sketch the graph:
x
y
– 2 2
4
2
f(x) = (1/2)x
14. Examples
Sketch the graph of the exponential function f(x) = (1/2)x.
Solution
Note the symmetry between the two functions:
x
y
– 2 2
4
2
f(x) = (1/2)x
f(x) = 2x
15. Properties of Exponential Functions
The exponential function y = bx (b > 0, b ≠ 1) has
the following properties:
1. Its domain is (– , ).
2. Its range is (0, ).
3. Its graph passes through the point (0, 1)
4. It is continuous on (– , ).
5. It is increasing on (– , ) if b > 1 and
decreasing on (– , ) if b < 1.
16. The Base e
Exponential functions to the base e, where e is an
irrational number whose value is 2.7182818…, play an
important role in both theoretical and applied problems.
It can be shown that
1
lim 1
m
m
e
m
17. Examples
Sketch the graph of the exponential function f(x) = ex.
Solution
Since ex > 0 it follows that the graph of y = ex is similar to the
graph of y = 2x.
Consider a few values for x:
x – 3 – 2 – 1 0 1 2 3
y 0.05 0.14 0.37 1 2.72 7.39 20.09
18. 5
3
1
Examples
Sketch the graph of the exponential function f(x) = ex.
Solution
Sketching the graph:
x
y
– 3 – 1 1 3
f(x) = ex
19. Examples
Sketch the graph of the exponential function f(x) = e–x
.
Solution
Since e–x
> 0 it follows that 0 < 1/e < 1 and so
f(x) = e–x
= 1/ex
= (1/e)x
is an exponential function with
base less than 1.
Therefore, it has a graph similar to that of y = (1/2)x
.
Consider a few values for x:
x – 3 – 2 – 1 0 1 2 3
y 20.09 7.39 2.72 1 0.37 0.14 0.05
20. 5
3
1
Examples
Sketch the graph of the exponential function f(x) = e–x
.
Solution
Sketching the graph:
x
y
– 3 – 1 1 3
f(x) = e–x
22. Logarithms
We’ve discussed exponential equations of the form
y = bx (b > 0, b ≠ 1)
But what about solving the same equation for y?
You may recall that y is called the logarithm of x to the
base b, and is denoted logbx.
✦ Logarithm of x to the base b
y = logbx if and only if x = by (x > 0)
23. Examples
Solve log3x = 4 for x:
Solution
By definition, log3x = 4 implies x = 34 = 81.
24. Examples
Solve log164 = x for x:
Solution
log164 = x is equivalent to 4 = 16x = (42)x = 42x, or 41 = 42x,
from which we deduce that
2 1
1
2
x
x
25. Examples
Solve logx8 = 3 for x:
Solution
By definition, we see that logx8 = 3 is equivalent to
3 3
8 2
2
x
x
27. Laws of Logarithms
If m and n are positive numbers, then
1.
2.
3.
4.
5.
log log log
b b b
mn m n
log log log
b b b
m
m n
n
log log
n
b b
m n m
log 1 0
b
log 1
b b
28. Examples
Given that log 2 ≈ 0.3010, log 3 ≈ 0.4771, and log 5 ≈ 0.6990,
use the laws of logarithms to find
log15 log3 5
log3 log5
0.4771 0.6990
1.1761
29. Examples
Given that log 2 ≈ 0.3010, log 3 ≈ 0.4771, and log 5 ≈ 0.6990,
use the laws of logarithms to find
log7.5 log(15 / 2)
log(3 5 / 2)
log3 log5 log2
0.4771 0.6990 0.3010
0.8751
30. Examples
Given that log 2 ≈ 0.3010, log 3 ≈ 0.4771, and log 5 ≈ 0.6990,
use the laws of logarithms to find
log81 4
log3
4log3
4(0.4771)
1.9084
31. Examples
Given that log 2 ≈ 0.3010, log 3 ≈ 0.4771, and log 5 ≈ 0.6990,
use the laws of logarithms to find
log50 log5 10
log5 log10
0.6990 1
1.6990
32. Examples
Expand and simplify the expression:
2 3
3
log x y 2 3
3 3
3 3
log log
2log 3log
x y
x y
33. Examples
Expand and simplify the expression:
2
2
1
log
2x
x
2
2 2
2
2 2
2
2
log 1 log 2
log 1 log 2
log 1
x
x
x x
x x
34. Examples
Expand and simplify the expression:
2 2
1
ln x
x x
e
2 2 1/2
2 2 1/2
2
2
( 1)
ln
ln ln( 1) ln
1
2ln ln( 1) ln
2
1
2ln ln( 1)
2
x
x
x x
e
x x e
x x x e
x x x
35. Examples
Use the properties of logarithms to solve the equation for x:
3 3
log ( 1) log ( 1) 1
x x
3
1
log 1
1
x
x
1
1
3 3
1
x
x
1 3( 1)
x x
1 3 3
x x
4 2x
2
x
Law 2
Definition of
logarithms
36. Examples
Use the properties of logarithms to solve the equation for x:
log log(2 1) log6
x x
log log(2 1) log6 0
x x
(2 1)
log 0
6
x x
0
(2 1)
10 1
6
x x
(2 1) 6
x x
2
2 6 0
x x
(2 3)( 2) 0
x x
2
x
Laws 1 and 2
Definition of
logarithms
3
2
log
x
x
is out of
the domain of ,
so it is discarded.
37. Logarithmic Function
The function defined by
is called the logarithmic function with base b.
The domain of f is the set of all positive numbers.
( ) log ( 0, 1)
b
f x x b b
38. Properties of Logarithmic Functions
The logarithmic function
y = logbx (b > 0, b ≠ 1)
has the following properties:
1. Its domain is (0, ).
2. Its range is (– , ).
3. Its graph passes through the point (1, 0).
4. It is continuous on (0, ).
5. It is increasing on (0, ) if b > 1
and decreasing on (0, ) if b < 1.
39. Example
Sketch the graph of the function y = ln x.
Solution
We first sketch the graph of y = ex.
1
x
y
1
y = ex
y = ln x
y = x
The required graph is
the mirror image of the
graph of y = ex with
respect to the line y = x:
40. Properties Relating
Exponential and Logarithmic Functions
Properties relating ex and ln x:
eln x = x (x > 0)
ln ex = x (for any real number x)
41. Examples
Solve the equation 2ex + 2 = 5.
Solution
Divide both sides of the equation by 2 to obtain:
Take the natural logarithm of each side of the equation
and solve:
2 5
2.5
2
x
e
2
ln ln2.5
( 2)ln ln2.5
2 ln2.5
2 ln2.5
1.08
x
e
x e
x
x
x
42. Examples
Solve the equation 5 ln x + 3 = 0.
Solution
Add – 3 to both sides of the equation and then divide both
sides of the equation by 5 to obtain:
and so:
5ln 3
3
ln 0.6
5
x
x
ln 0.6
0.6
0.55
x
e e
x e
x
44. Applied Example: Growth of Bacteria
In a laboratory, the number of bacteria in a culture grows
according to
where Q0 denotes the number of bacteria initially present
in the culture, k is a constant determined by the strain of
bacteria under consideration, and t is the elapsed time
measured in hours.
Suppose 10,000 bacteria are present initially in the culture
and 60,000 present two hours later.
How many bacteria will there be in the culture at the end
of four hours?
0
( ) kt
Q t Q e
45. Applied Example: Growth of Bacteria
Solution
We are given that Q(0) = Q0 = 10,000, so Q(t) = 10,000ekt.
At t = 2 there are 60,000 bacteria, so Q(2) = 60,000, thus:
Taking the natural logarithm on both sides we get:
So, the number of bacteria present at any time t is given by:
0
2
2
( )
60,000 10,000
6
kt
k
k
Q t Q e
e
e
2
ln ln 6
2 ln 6
0.8959
k
e
k
k
0.8959
( ) 10,000 t
Q t e
46. Applied Example: Growth of Bacteria
Solution
At the end of four hours (t = 4), there will be
or 360,029 bacteria.
0.8959(4)
(4) 10,000
360,029
Q e
47. Applied Example: Radioactive Decay
Radioactive substances decay exponentially.
For example, the amount of radium present at any time t
obeys the law
where Q0 is the initial amount present and k is a suitable
positive constant.
The half-life of a radioactive substance is the time required
for a given amount to be reduced by one-half.
The half-life of radium is approximately 1600 years.
Suppose initially there are 200 milligrams of pure radium.
a. Find the amount left after t years.
b. What is the amount after 800 years?
0
( ) (0 )
kt
Q t Q e t
48. Applied Example: Radioactive Decay
Solution
a. Find the amount left after t years.
The initial amount is 200 milligrams, so Q(0) = Q0 = 200, so
Q(t) = 200e–kt
The half-life of radium is 1600 years, so Q(1600) = 100, thus
1600
1600
100 200
1
2
k
k
e
e
49. Applied Example: Radioactive Decay
Solution
a. Find the amount left after t years.
Taking the natural logarithm on both sides yields:
Therefore, the amount of radium left after t years is:
1600 1
ln ln
2
1
1600 ln ln
2
1
1600 ln
2
1 1
ln 0.0004332
1600 2
k
e
k e
k
k
0.0004332
( ) 200 t
Q t e
50. Applied Example: Radioactive Decay
Solution
b. What is the amount after 800 years?
In particular, the amount of radium left after 800 years is:
or approximately 141 milligrams.
0.0004332(800)
(800) 200
141.42
Q e
51. Applied Example: Assembly Time
The Camera Division of Eastman Optical produces a single
lens reflex camera.
Eastman’s training department determines that after
completing the basic training program, a new, previously
inexperienced employee will be able to assemble
model F cameras per day, t months after the employee starts
work on the assembly line.
a. How many model F cameras can a new employee assemble
per day after basic training?
b. How many model F cameras can an employee with one
month of experience assemble per day?
c. How many model F cameras can the average experienced
employee assemble per day?
0.5
( ) 50 30 t
Q t e
52. Applied Example: Assembly Time
Solution
a. The number of model F cameras a new employee can
assemble is given by
b. The number of model F cameras that an employee with
1, 2, and 6 months of experience can assemble per day is
given by
or about 32 cameras per day.
c. As t increases without bound, Q(t) approaches 50.
Hence, the average experienced employee can be expected
to assemble 50 model F cameras per day.
(0) 50 30 20
Q
0.5(1)
(1) 50 30 31.80
Q e