4.3 – Logarithmic Functions

1. 4.3 – Logarithmic Functions 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 • Solve2x=8 using mental math. • Now solve 2x=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. 1 6 1 2 Example 1 Write each exponential equation in logarithmic form.

6. 1 3 Example 2 Write each logarithmic form in exponential equation. 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 = log105. • 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 = 2x, is a logarithmic function, such as y = log2x. • You may notice that the domain and range of each function are switched. • The domain of y = 2x is all real numbers (R), and the range is {y|y > 0}. The domain of y = log2x is {x|x > 0}, and the range is all real numbers (R).

11. x –2 –1 0 1 2 f(x) = 1.25x 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) = 1.25x • Graph f(x) = 1.25x by using a table of values.

12. x f–1(x) = log1.25x Example 5 continued… • Graph the inverse, f–1(x) = log1.25x,by using a table of values. • Domain: • Range:

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 1 2

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.

15. Calculating Logarithms other than base 10 • Example 8: log7 343 = • Example 9: log3 ( ) =

16. Your turn… • Example 10: log.5 .25 =

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