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Section 4.6 Logarithmic Functions. Objectives: 1. To convert between exponential and logarithmic form. 2. To evaluate logarithms of numbers. 3. To graph logarithmic functions and give their domains and ranges.

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## Section 4.6 Logarithmic Functions

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**Section 4.6**Logarithmic Functions**Objectives:**1. To convert between exponential and logarithmic form. 2. To evaluate logarithms of numbers. 3. To graph logarithmic functions and give their domains and ranges.**The inverse of the exponential function is the logarithmic**function.**f(x) = 2x**y = x f(x) = log2x**Definition**Logarithmic function f(x) = logax, where a > 0, a 1, and x > 0. It is the inverse of the exponential function f(x) = ax.**The rule y = logax is equivalent to ay = x; therefore, a is**the base of the logarithm and y, the logarithm, is an exponent.**The logarithmic expression can be written in either of two**forms. Log Form Exponential Form logax = y ↔ay = x**EXAMPLE 1Write log4 64 = y in exponential form.**log4 64 = y 4y = 64**EXAMPLE 2Write 3-5 = in logarithmic form.**3-5 = 1 243 1 243 1 243 log3 = -5**EXAMPLE 3Prove loga 1 = 0 a 0 with a 1.**a0 = 1 loga 1 = 0**A logarithm in base ten is called a common logarithm. The**base of a common log is not usually written. Common logs can be found using the log key on your calculator.**The second special type of logarithm is the natural**logarithm whose base is e. Recall that e is an irrational number that is approximately 2.71828. The special notation for a natural logarithm is ln.**n**æ 1 ö ç ÷ e = lim 1 + n è ø n 1 1 1 1 ¥ å + + + e = = 1 + 1 + ∙∙∙ n! 2 6 24 n=0 e, also called the Euler number, is defined as or**Homework**pp. 202-203**►A. Exercises**Change the following logarithms to exponential form. 1. log2x = y**1**16 ►A. Exercises Change the following logarithms to exponential form. 3. log4 = -2**►A. Exercises**Change the following logarithms to exponential form. 5. log 1000 = 3**►A. Exercises**Change the following logarithms to exponential form. 7. ln 1 = 0**►A. Exercises**Explain why the following are true. 9. loga a = 1**►A. Exercises**Change the following exponential expressions to log form. 11. 53 = 125**►A. Exercises**Change the following exponential expressions to log form. 13. 82 = 64**►A. Exercises**Change the following exponential expressions to log form. 15. 73 = 343**►B. Exercises**Graph. Give the domain and range of each. 17. y = log4x**►B. Exercises**Graph. Give the domain and range of each. 17. y = log4x**►B. Exercises**Evaluate. 21. log5 1,953,125 5y = 1,953,125 5y = 59 y = 9**►B. Exercises**Evaluate. 23. log2 1 8 2y = 8-1 2y = (23)-1 2y = 2-3 y = -3**►B. Exercises**Evaluate. 25. log6 1 1296**■Cumulative Review**Use interval notation to show the intervals of continuity of the following functions. 29. f(x) = 3x2 – 5 x**■Cumulative Review**Use interval notation to show the intervals of continuity of the following functions. 30. g(x) = 3x2 – 7x + 9**■Cumulative Review**Use interval notation to show the intervals of continuity of the following functions. 31. h(x) = x – 3**■Cumulative Review**32. Name three characteristics of a graph that cause a function to be discontinuous. Name an example of each.**■Cumulative Review**33. Certain functions, such as the absolute value function, do not graph as a smooth curve, but have sharp turns in their graphs. Is such a function discontinuous?

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