1. SECTION 4.4 LOGARITHMIC FUNCTIONS LOGARITHMIC FUNCTIONS

2. LOGARITHMIC FUNCTIONS The logarithm (base b) of a number is the power to which b must be raised to get that number.

3. EXAMPLES: (a) log 4 16 = 2 (b) log 3 27 = 3 (c) log e e 4 = 4 (d) log 2 32 = 5 (e) log 5 1 = 0 (f) log 2 1/8 = - 3 (g) log 10.1 = - 1 (h) log 9 27 = 3/2

4. LOGARITHMIC FUNCTIONS Recall that only one-to-one functions have an inverse. Exponential functions are one-to-one. Their inverses are logarithmic functions.

5. LOGARITHMIC FUNCTIONS Example: Change the exponential expressions to logarithmic expressions. 1.2 3 = m e b = 9 a 4 = 24

6. LOGARITHMIC FUNCTIONS Example: Change the logarithmic expressions to exponential expressions. log a 4 = 5 log e b = - 3 log 3 5 = c

7. DOMAIN OF A LOGARITHMIC FUNCTION Since the logarithmic function is the inverse of the exponential, the domain of a logarithmic is the same as the range of the exponential.

8. DOMAIN OF A LOGARITHMIC FUNCTION Example: Find the domain of the functions below: F(x) = log 2 (1 - x)

9. SPECIAL LOGARITHMS COMMONLOGARITHM NATURALLOGARITHM Logarithm to the base 10. Ex:log 100 = 2 Logarithm to the base e. Ex:ln e 2 = 2

10. THE NATURAL LOGARITHMIC FUNCTION Graph the function g(x) = lnx in the same coordinate plane with f(x) = e x Graph the function g(x) = lnx in the same coordinate plane with f(x) = e x Notice the symmetry with respect to the line y = x. Notice the symmetry with respect to the line y = x.

11. f(x) = e x g(x) = lnx Compose the two functions: g(f(x)) = ln e x = x f(g(x)) = e ln x = x We can see graphically as well as algebraically that these two functions are inverses of each other.

12. Given f(x) = b x Then f -1 (x) = log b x

13. GRAPHS OF LOGARMITHMIC FUNCTIONS 1.The x-intercept is 1. 2.The y-axis is a vertical asymptote of the graph. 3.A logarithmic function is decreasing if 0 1. 4.The graph is continuous.

14. GRAPHING LOGARITHMIC FUNCTIONS USING TRANSFORMATIONS Graph f(x) = 3log(x – 1). Determine the domain, range, and vertical asymptote of f.

15. EXAMPLE Graph the function f(x) = ln(1 - x). Determine the domain, range, and vertical asymptote.

16. SOLVING A LOGARITHMIC EQUATION Solve: log 3 (4x – 7) = 2 Solve: log x 64 = 2

17. USING LOGARITHMS TO SOLVE EXPONENTIAL EQUATIONS Solve: e 2x = 5

18. EXAMPLE DO EXAMPLE 10 ON ALCOHOL AND DRIVING

19. CONCLUSION OF SECTION 4.4 CONCLUSION OF SECTION 4.4