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- 1. EXPONENTIAL AND LOGARITHMIC FUNCTIONS <br />
- 2. EXPONENTIAL FUNCTION<br />If x and b are real numbers such that b > 0 and b ≠ 1, then f(x) = bx is an exponential function with base b.<br />Examples of exponential functions:<br /> a) y = 3x b) f(x) = 6x c) y = 2x<br />Example: Evaluate the function y = 4xat the given values of x.<br /> a) x = 2 b) x = -3 c) x = 0 <br />
- 3. PROPERTIES OF EXPONENTIAL FUNCTION y = bx<br /><ul><li> The domain is the set of all real numbers.
- 4. The range is the set of positive real numbers.
- 5. The y – intercept of the graph is 1.
- 6. The x – axis is an asymptote of the graph.
- 7. The function is one – to – one. </li></li></ul><li>The graph of the function y = bx<br />y<br />1<br />x<br />o<br />
- 8. EXAMPLE 1: Graph the function y = 3x<br />y<br />1<br />x<br />o<br />
- 9. EXAMPLE 2: Graph the function y = (1/3)x<br />y<br />1<br />x<br />o<br />
- 10. NATURAL EXPONENTIAL FUNCTION: f(x) = ex<br />y<br />1<br />x<br />o<br />
- 11. LOGARITHMIC FUNCTION<br />For all positive real numbers x and b, b ≠ 1, the inverse of the exponential function y = bx is the logarithmic functiony = logbx.<br />In symbol, y = logb x if and only if x = by<br />Examples of logarithmic functions:<br /> a) y = log3 x b) f(x) = log6 x c) y = log2 x <br />
- 12. EXAMPLE 1: Express in exponential form:<br />EXAMPLE 2: Express in logarithmic form:<br />
- 13. PROPERTIES OF LOGARITHMIC FUNCTIONS<br /><ul><li> The domain is the set of positive real numbers.
- 14. The range is the set of all real numbers.
- 15. The x – intercept of the graph is 1.
- 16. The y – axis is an asymptote of the graph.
- 17. The function is one – to – one. </li></li></ul><li>The graph of the function y = logb x<br />y<br />x<br />o<br />1<br />
- 18. EXAMPLE 1: Graph the function y = log3 x<br />y<br />1<br />x<br />o<br />
- 19. EXAMPLE 2: Graph the function y = log1/3 x<br />y<br />x<br />1<br />o<br />
- 20. PROPERTIES OF EXPONENTS<br />If a and b are positive real numbers, and m and n are rational numbers, then the following properties holds true:<br />
- 21. To solve exponential equations, the following property can be used:<br />bm = bn if and only if m = n and b > 0, b ≠ 1<br />EXAMPLE 1: Simplify the following:<br />EXAMPLE 2: Solve for x:<br />
- 22. PROPERTIES OF LOGARITHMS<br />If M, N, and b (b ≠ 1) are positive real numbers, and r is any real number, then <br />
- 23. Since logarithmic function is continuous and one-to-one, every positive real number has a unique logarithm to the base b. Therefore,<br />logbN = logbM if and only if N = M <br />EXAMPLE 1: Express the ff. in expanded form: <br />
- 24. EXAMPLE 2: Express as a single logarithm:<br />
- 25. NATURAL LOGARITHM<br />Natural logarithms are to the base e, while common logarithms are to the base 10. The symbol ln x is used for natural logarithms.<br />EXAMPLE: Solve for x:<br />
- 26. CHANGE-OF-BASE FORMULA <br />EXAMPLE: Use common logarithms and natural logarithms to find each logarithm:<br />
- 27. Solving Exponential Equations<br />Guidelines:<br />1. Isolate the exponential expression on one side of the equation.<br />2. Take the logarithm of each side, then use the law of logarithm to bring down the exponent.<br />3. Solve for the variable.<br />EXAMPLE: Solve for x:<br />
- 28. Solving Logarithmic Equations<br />Guidelines:<br />1. Isolate the logarithmic term on one side of the equation; you may first need to combine the logarithmic terms.<br />2. Write the equation in exponential form.<br />3. Solve for the variable.<br />EXAMPLE 1: Solve the following:<br />
- 29. EXAMPLE: Solve for x:<br />
- 30. Application: (Exponential and Logarithmic Equations)<br />The growth rate for a particular bacterial culture can be calculated using the formula B = 900(2)t/50, where B is the number of bacteria and t is the elapsed time in hours. How many bacteria will be present after 5 hours?<br />How many hours will it take for there to be 18,000 bacteria present in the culture in example (1)?<br />A fossil that originally contained 100 mg of carbon-14 now contains 75 mg of the isotope. Determine the approximate age of the fossil, to the nearest 100 years, if the half-life of carbon-14 is 5,570 years. <br />
- 31. In a town of 15,000 people, the spread of a rumor that the local transit company would go on strike was such that t hours after the rumor started, f(t) persons heard the rumor, where experience over time has shown that<br /> a) How many people started the rumor?<br /> b) How many people heard the rumor after 5 hours?<br />5. A sum of $5,000 is invested at an interest rate of 5% per year. Find the time required for the money to double if the interest is compounded (a) semi-annually (b) continuously.<br />

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