1. Objective: Students will be able to write equivalent forms for exponential and logarithmic functions, and can write, evaluate, and graph logarithmic functions.

2. Vocabulary Logarithm – the exponent to which a specified base is raised to obtain a given value. It is the inverse of an exponent. Common logarithm – a logarithm with base 10 Logarithmic function – the inverse of an exponential function

3. Finding Logarithms Solve 2 x =8 using mental math. Now solve 2 x =512 This problem would be much easier to solve if you could do so by taking the “mental math” out. This inverse operation is called finding a logarithm.

4. From exponential to logarithmic Form You can write an exponential equation as a logarithmic equation and vice versa.

5. Example 1 Write each exponential equation in logarithmic form. Exponential Equation Logarithmic Form 3 5 = 243 25 = 5 10 4 = 10,000 6 –1 = a b = c 1 2 1 6

6. Example 2 Write each logarithmic form in exponential equation. Logarithmic Form Exponential Equation log 9 9 = 1 log 2 512 = 9 log 8 2 = log 4 = –2 log b 1 = 0 1 3 1 16

7. A logarithm is an exponent, so the rules for exponents also apply to logarithms.

8. Common Logarithms A logarithm with base 10 is called a common logarithm. If no base is written for a logarithm, the base is assumed to be 10. For example, log 5 = log 10 5. Example 3: Evaluate each log in your calculator. log 0.01

9. Example 4 log 1000

10. Logarithmic Functions Because logarithms are the inverses of exponents, the inverse of an exponential function, such as y = 2 x, is a logarithmic function, such as y = log 2 x. You may notice that the domain and range of each function are switched. The domain of y = 2 x is all real numbers ( R ), and the range is {y|y > 0}. The domain of y = log 2 x is {x|x > 0}, and the range is all real numbers ( R ).

11. Graphing Logarithmic Functions Example 5: Use the x-values {–2, –1, 0, 1, 2}. Graph the function and its inverse. Describe the domain and range of the inverse function. f(x) = 1.25 x Graph f(x) = 1.25 x by using a table of values. f(x) = 1.25 x 210–1–2x

12. Example 5 continued… Graph the inverse, f –1 (x) = log 1.25 x, by using a table of values. Domain: Range: f –1 (x) = log 1.25 x x

13. Example 6 Use the x-values {–2, –1, 0, 1, 2}. Graph the function and its inverse. Describe the domain and range of the inverse function. f(x) = ( ) x 2 1 x–2–1012 f(x) =( ) x x421 f –1 (x) =log x –2–1012

14. Your turn… Example 7: Use x = –2, –1, 1, 2, and 3 to graph f(x) = (.75) x Then graph its inverse. Describe the domain and range of the inverse function. x–2–1123 f(x) = x x f –1 (x) = log x –2–1123

15. Calculating Logarithms other than base 10 Example 8: log 7 343 = Example 9: log 3 ( ) =

16. Your turn… Example 10: log.5.25 =

17. Homework for tonight Homework # ____ Textbook pg. 253 # 18 – 26 even, 29, 30