Section 5.5 properties of logarithms
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This document contains examples of properties of logarithms, including: 1) Writing logarithms with the same base as a sum or difference of logarithms by using properties 1-4 2) Expressing logarithms with different bases in terms of a single logarithm using properties 5-6 3) Approximating logarithms by changing them to an exponential form and using properties 5 and 7 The document provides step-by-step workings for each example and identifies the relevant property of logarithms used at each step.
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3. The product and quotient laws state that for logarithms with the same base, log(ab) = log(a) + log(b) and log(a/b) = log(a) - log(b).
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This document discusses evaluating and graphing logarithmic functions. It defines logarithms and explains how to evaluate logarithmic expressions without a calculator by rewriting them as exponential expressions. Examples are provided such as log 3 81 = 4 and log 2 (1/32) = -5. The definitions of common (base 10), natural (base e), and general logarithmic functions are given. Graphs of logarithmic functions are asymptotic to the horizontal axis and shift right or left depending on the base.
鳳山高級中學 B1 3 3---ans
This document contains a math exam with multiple choice and calculation questions. It provides the questions, solutions, and explanations. The questions cover topics like logarithms, coordinate planes, and functions involving logarithms. The final question asks to determine the values of a and b given a function f(x) = b + loga x and the function values at four points. The solution is that a = 2 and b = 5.
MATHS SYMBOLS - EXPONENTIALS + LOGARITHMS and THEIR PROPERTIES
MATHS SYMBOLS - EXPONENTIALS + LOGARITHMS and THEIR PROPERTIESIst. Superiore Marini-Gioia - Enzo Exposyto
MATHS SYMBOLS - EXPONENTIALS + LOGARITHMS and THEIR PROPERTIES - DEFINITIONS - 10+ PROPERTIES - ANTILOGARITHMS - INVERSE OPERATIONS - NATURAL LOGARITHMS - NEPER - NAPIER - EULER'S NUMBER - LOG - LN - POWER - ROOT - PRODUCT - QUOTIENT - CHANGE of BASE - PROOFS, EXAMPLES and CALCULATIONS STEP by STEP Lecture 11 sections 4.3-4.4 logarithmic functions
This document discusses logarithmic functions and their properties. It begins with examples of converting between exponential and logarithmic form. It then covers the natural logarithm function and using transformations of logarithmic functions. The main section discusses properties of logarithms, including the product rule, quotient rule, and power rule for logarithms. It provides examples of writing logarithmic expressions using these rules in both expanded and condensed form.
5indiceslogarithms 120909011915-phpapp02
Logarithms and indices are important mathematical concepts. Laws and formulas allow logarithmic and index expressions to be simplified and equations to be solved.
Key steps in working with logarithms and indices include using the appropriate laws and formulas to simplify expressions, setting up logarithmic or index equations equal to each other, and solving the resulting linear equations.
Practice questions cover a range of skills like evaluating logarithmic expressions without a calculator, expressing logarithmic values in terms of given variables, and solving equations involving logarithms and indices. Mastering the laws, formulas, and problem-solving process is essential for working with logarithms and indices.
Week 7 Forum All problems based on material posted in .docx
Week 7 Forum
All problems based on material posted in Content – Course Resources – eReading
section for week 7. Study this material before you post your submission.
Remember: log means logarithm with base 10: log = log10
and ln means natural logarithm with base e(2.718) ln = loge
Last Name
Problem-1.
Use the property of logarithms
to write it as a single logarithm.
Problem-2.
Solve the given exponential or
logarithmic equation.
1. Bah Bello 2ln(x) +ln(y) – 4ln(z) 3
2x + 5
= 81
2. Buckson log2(x) – log2(y) – 3log2(z) log(2x + 30) = 2
3. Cuevas 2log(x) – ½log(y) + 3log(z) 25
x –1
= 5
4. Davis log3(x) + log3(y) – 2log3(z) log5(10 – x) = 2
5. Headen-Vance log5(x) – 2log5(y) + 3log5(z) 2
x + 4
= 16
6. Hicks ln(20x) – ln(4) – ln(5) 10
2x – 5
= 0.1
7. Hugginscrawford ½ ln(x) – ½ ln(y) + ½ ln(z) log2(x - 12) = 3
8. Ioannides log(x) + ½log(y) – 2log(z) 34x – 6 =
9. Joseph ln(100x) – ln(10y) – ln(5) 5
2x –1
= 125
10. Keyser log(x) + 2log(y) – 1 log(x - 5) + log(x + 5) = 2
11. Linne ½ ln(10) + ½ ln(x) – ½ ln(5) 25x – 4 =
12. Madison log(4x) + log(5y) – log(20) 5
2x
= 8
13. Manhertz 4log(x) + ½log(y) – 3log(z) 10
x + 3
= 100
14. McCoy-Smith ½ log(x) – log(y) + 2log(z) log2(x - 4) + log2(x + 4) = 3
15. Merideth ½ log(x) – log(y) – 4log(z) 52x – 6 =
16. Nell ln(12x) + ln(5) – ln(6) 10
2x – 4
= 0.01
17. Nieten ½ ln(20) – ½ ln(10) + ½ ln(x) log3(x - 2) + log3(x + 2) = 1
18. Pryce log2(x) + 3log2(y) + 4 8
x – 10
= 11
19. Pugh log(20x) + log(5y) – log(25) 3
x + 5
= 7
20. Royster 2 – log(x) – log(y) 4x – 5 =
21. Salomon log(15) + log(2) – 1 7
x - 5
= 49
22. Schultz ln(20x) + ln(3) – ln(15) log(x - 10) + log(x + 10) =2
23. Sidhwani 2ln(x) + ln(y) – ½ ln(2) 4
x - 3
= 2
24. Smith log5(x) – 2log5(y) + 2 3
x – 10
=
25. Thousand 2ln(x) + 3ln(y) – 4ln(z) ln(x+1) – ln(x) = 1
26. Tranchilla ½ ln(x) + ln(y) – 3ln(z) log(x+4) – log(x) = 1
27. Vasquez ln(x) + ½ln(y) – ½ln(z) 3
5x – 2
= 1
28. Waugaman 4log(x) – 3log(y) – 4log(z) log(x
2
– 3x) = 1
29. Westfall 2log2(x) – log2(y) + 3 2
4x – 5
=
30. Williams 2 – log2(x) – log2(y) log5(2x - 15) = 3
31. Wolod log(4x) + log(3y) – log(6) 5
4x - 3
= 125
...
P7
This document provides a summary of key concepts from a college algebra textbook, including:
Rational expressions involve fractions of polynomials. The domain of an algebraic expression is the set of real numbers for which the expression is defined. Compound fractions contain fractions in the numerator, denominator, or both. Rational expressions can be simplified by factoring and canceling common factors. Adding and subtracting rational expressions requires finding a common denominator. The denominator of a fraction can be rationalized by multiplying the numerator and denominator by the conjugate radical. Common errors involve applying properties of multiplication to addition incorrectly.
MATHS SYMBOLS - #3 - LOGARITHMS - DEFINITION - EXAMPLES
MATHS SYMBOLS
#3 - LOGARITHMS
DEFINITION of LOGARITHM
EXAMPLES
ANTILOGARITHMS
INVERSE OPERATIONS
NATURAL LOGARITHMS
NEPER - NAPIER - EULER'S NUMBER
LOG - LN
POWER - ROOT
PRODUCT - QUOTIENT
CHANGE of BASE
PROOFS - EXAMPLES
CALCULATIONS STEP by STEP
www.enzoexposito.it/mobile/matematica.html
www.enzoexposito.it/mobile/expon_log.html
C5-Logarithms.pdf
This document defines logarithmic functions and explores their properties. It contains the following key points:
1. Logarithms are defined as the exponent to which a base number must be raised to equal the value. Common and natural logarithms use bases of 10 and e respectively.
2. Logarithmic and exponential functions are inverses. Basic properties include logb1=0, logbbx=x, and blogbx=x.
3. The laws of logarithms allow logarithmic expressions to be condensed using logb(uv)=logbu+logbv, logb(u/v)=logbu-logbv, and logbun=nlogbu.
Alg2 lesson 10-4 and 10-5
A common logarithm has a base of 10. The log key on a calculator finds common logs. Exponential and logarithmic functions can be used to describe quantities that grow or decay continuously over time.
0404 ch 4 day 4
The document discusses logarithmic functions and their properties. It begins by explaining that the logarithmic function is the inverse of the exponential function. It then provides examples of finding the inverse of exponential functions and graphing logarithmic functions. The rest of the document focuses on working with logarithms, including: the common bases of base 10 (logarithm) and base e (natural logarithm); converting between exponential and logarithmic forms; and using the change of base formula to evaluate logarithms with bases other than 10 or e.
Business Math Chapter 2
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.
0405 ch 4 day 5
The document discusses logarithmic functions and graphs logarithmic equations, exploring their asymptotes, intercepts, domains, and ranges. It also covers properties of natural logarithms, including that ln(1)=0, ln(e)=1, and that logarithmic and exponential functions are inverses of each other.
Questions and Solutions Logarithm.pdf
Unlock a deep understanding of mathematics with our Module and Summary! Clear definitions, comprehensive discussions, relevant example problems, and step-by-step solutions will guide you through mathematical concepts effortlessly. Learn with a systematic approach and discover the magic in every step of your learning journey. Mathematics doesn't have to be complicated—let's make it simple and enjoyable!
Section 6.2 trigonometric functions unit circle approach
This document section discusses trigonometric functions using the unit circle approach. It defines trigonometric functions of angles based on the coordinates of points on the unit circle. Several examples are worked out, finding exact values of trigonometric functions like cosine, tangent, and sine at specific angles by considering the appropriate points on the unit circle. It also discusses using identities to find values of other trig functions in terms of sine and cosine.
factoring and the other ones polynomials2.ppt
The document discusses various methods for factoring polynomials:
1) Factoring out the greatest common factor (GCF) of each term.
2) Using the difference of squares formula a2 - b2 = (a + b)(a - b).
3) Grouping terms to produce common binomial factors.
4) Reversing the FOIL process to factor trinomials of the form x2 + bx + c.
5) Factoring polynomials completely by factoring out the GCF and then factoring the remaining terms.
4.6 more on log and exponential equations t
This document contains examples of solving various logarithmic and exponential equations without using a calculator:
Example A solves the equation 2*42x – 1 = 81 – 3x by rewriting it in base 2 and then solving for x.
Example B combines logarithms using the product rule to solve log2(x – 1) + log2(x + 3) = 5 for x.
Example C combines logarithms using the quotient rule to solve log4(x + 2) – log4(x – 1) = log4 (5) for x.
Example D uses a base change formula to isolate the exponential part and solve 14 = 3·52x+1 for x by re
EPCA_MODULE-2.pptx
To factor a polynomial using greatest common factors (GCF):
1. Find the GCF of the coefficients and of the variables.
2. The GCF is the factored form of the polynomial.
3. Checking the factored form using the distributive property verifies the correct factorization.
Similar to Section 5.5 properties of logarithms (20)
Special topics about stocks and bonds using algebra
Special topics about stocks and bonds using algebra
MATHS SYMBOLS - EXPONENTIALS + LOGARITHMS and THEIR PROPERTIES
MATHS SYMBOLS - EXPONENTIALS + LOGARITHMS and THEIR PROPERTIES
Week 7 Forum All problems based on material posted in .docx
Week 7 Forum All problems based on material posted in .docx
MATHS SYMBOLS - #3 - LOGARITHMS - DEFINITION - EXAMPLES
MATHS SYMBOLS - #3 - LOGARITHMS - DEFINITION - EXAMPLES
Section 6.2 trigonometric functions unit circle approach
Section 6.2 trigonometric functions unit circle approach
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Section 5.5 properties of logarithms
- 1. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Section 5.5 Properties of Logarithms
- 2. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
- 3. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
- 4. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
- 5. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 3 (a) log 3π π = 5log 3 (b) 5 3= (c) lne0.35t = 0.35t Property (2) Property (1) Property (2)
- 6. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
- 7. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
- 8. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. ( )2 3 2Write log 1 , >1, as a sum of logarithms. Express all powers as factors. x x x− 2 3 2 2log log 1x x+ − ( ) 1 2 3 2 2log log 1x x= + − ( )2 2 1 2log log 1 3 x x= + −
- 9. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. ( ) 4 6 22 Write log , 0, as a difference of logarithms. 3 Express all powers as factors. x x x ≠ + ( ) 2 4 2 6 6log log 3x x− + ( )2 6 64log 2log 3x x− +
- 10. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. ( ) 3 2 2 Write ln , 2, as a sum and difference of logarithms. 1 Express all powers as factors. x x x x − > + ( ) 23 ln 2 ln 1x x x− − + ( ) ( ) 1 23 2ln ln 2 ln 1x x x+ − − + ( ) ( ) 1 3ln ln 2 2ln 1 2 x x x+ − − +
- 11. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
- 12. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. ( ) ( ) ( ) ( ) ( ) 2 2 Write each of the following as a single logarithm. a 3ln 2 ln 1 b log 4 2log 5 2 c 2log 3 3log 2 log 1 a a a a a x x + − − + − + ( )3 2 ln 2 ln x= + ( )2 ln 8x= 1 22 log 4 log 5a a= − 2 log 25 a = ÷ ( )2 3 2 log 3 log 2 log 1a a a x− = + − + ( )( ) ( )3 2 2 log 2 3 log 1a a x− = − + Property (5) Property (3) Property (5) Property (4) Property (5) Property (3) ( )2 8 log 9 1 a x ÷= ÷+ Property (4)
- 13. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
- 14. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
- 15. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 3Approximate log 12. Round answer to four decimal places. 3log 12y = 3 12y = Exponential form ln3 ln12y = Property (7) ln3 ln12y = Property (5) ln12 ln3 y = Exact value 2.2619y ≈ Approximate value
- 16. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
- 17. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
- 18. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.