Math 150 c Lynch 1 of 6 Section 7.2 Logarithmic Functions Definition. Let a be any positive number not equal to 1. The logarithm of x to the base a is y if and only if a y = x. The number y is denoted by y = log a x In other words, we say that y is the log of x to the base a, written y = log a x, if when we raise a to the y th power we get x. Note. The logarithm to a base a is the inverse function of a x. The expression log a x is read the log of x to the base a or the log to the base a of x. Remember, to evaluate log a x we can write it as an exponential: y = log a x if and only if a y = x Example 1. Evaluate the following logarithms by writing an exponential equation. (a) log 10 10 = because (f) log 5 25 = because (b) log 100 10 = because (c) log 10 10 = because (g) log 2 64 = because (h) log 4 64 = because (d) log 10 100 = because (i) log 8 64 = because (e) log 10 1000 = because Example 2. If (2.3) 4 = x, what is a in log a x = 4? Example 3. If the logarithm of x to the base 3 is 4, then x must equal what? Example 4. Solve the equation log 9 x = 3 for x. Example 5. If log a 36 = 2, then what is a?
Math 150 c Lynch 7.2 Logarithmic Functions 2 of 6 Theorem. The graph of any logarithmic function y = log a x has the following properties: The fucntion y = log a x has domain (0, ) and the range is (, ). Since a 0 = 1, then log a 1 = 0 and (1, 0) is on the graph. Since a 1 = a, then log a a = 1 and (a, 1) is on the graph. Since a 1 = 1 a, then log a ( 1 a) = 1 and ( 1 a, 1) is on the graph. The function has a vertical asymptote at x = 0 and no horizontal asymptote. For a > 1, the plot of the logarithmic function with base a, written y = log a x is increasing on its entire domain (0, ) and looks like the following: For 0 < a < 1, the plot of the logarithmic function with base a, written y = log a x is decreasing on its entire domain (0, ) and looks like the following: Example 6. Find the domain of the function f(x) = e 3x 2 log 3 (2x 5). Example 7. Graph the following functions. Give the domain, range, and intercepts of the function. (a) f(x) = log 7 x
Math 150 c Lynch 7.2 Logarithmic Functions 3 of 6 (b) g(x) = 2 log 3 (x 3) 3 (c) h(x) = 3 log 1 (x + 1) 3 Properties of Logarithms Theorem. Logarithms have the following properties (you need to memorize these): 1. The domain of log a x is (0, ) and its range is (, ). 2. a log a x = x. 3. log a (a x ) = x 4. log a x = log a y if and only if x = y 5. log a 1 = 0 6. log a (xy) = log a x + log a y 7. log a (x y ) = y log a x ( ) x 8. log a y = log a x log a y Example 8. Simplify the expressions: (a) 3 2 log 9 5 (b) 7 4 log 7 3
Math 150 c Lynch 7.2 Logarithmic Functions 4 of 6 Example 9. Simplify the following expressions: (a) log 4 7 3 log 4 3 + 2 log 4 5 (b) 2 log 6 10 + 3 log 6 3 log 6 75 Note. When solving equations with logarithms, we MUST CHECK that our solutions are in the domain of the original equation. Example 10. Solve the following expressions for x. (a) 3 + log 5 4 + 2 log 5 x = 10 (b) 7 2x+3 = 11 Inverses Theorem. The logarithmic function with base a and the exponential function with base a are inverses since log a x = y if and only if a y = x Also, note the composition of the logarithmic function and exponential function equal x: a log a x = x and log a a x = x Example 11. Graph the functions f(x) = 3 x and g(x) = log 3 x. Find the domain, range, and intercepts of each function.
Math 150 c Lynch 7.2 Logarithmic Functions 5 of 6 Example 12. If a 5 = 7.33, what is log a 7.33? Example 13. If log 5 13 = y, what is 5 y? Natural Logarithm Definition. The natural log function is the log function with base e. Instead of writing log e x, we write ln x. The natural log function ln x is the inverse of the natural exponential function e x. Therefore, ln e x = x and e ln x = x Example 14. Graph e x and ln x. functions. Give the domain, range, and intercepts of both Example 15. Solve the equation e 2x+7 = 11. Example 16. Suppose that f(x) = ae kx for some value of k and a. Suppose that f(1) = 2 and f(3) = 1. Find a and k.
Math 150 c Lynch 7.2 Logarithmic Functions 6 of 6 Change of Base Theorem. For any a, b > 0, we can rewrite the logarithmic function log a x using the logarithmic function with base b using the formula log a x = log b x log b a Also, the exponential function a x can be rewritten for base b as a x = b x log b a Note. When we change base, we usually change to base e. So we can rewrite a logarithmic function and exponential function as log a b = ln b ln a and a x = e x ln a Note. Most calculators only have a button for log 10 x and ln x. To evaluate a logarithmic function with any other base, we must use the change of base formula Example 17. Use a calculator, to evaluate log 13 25. Example 18. Suppose that ln 2 = a, ln 3 = b, ln 5 = c, ln 7 = d, and ln 11 = f. Evaluate and fully simplify the following, writing your answer in terms of a, b, c, d, and f. (a) 11 log 6 75 (b) 5 log 55 24.