1. 6. 3 Logarithmic Functions Objectives: Write equivalent forms for exponential and logarithmic equations. Use the definitions of exponential and logarithmic functions to solve equations. Standard: 2.8.11.S. Analyze properties and relationships of functions.

2. Logarithms are used to find unknown exponents in exponential models. Logarithmic functions define many measurement scales in the sciences, including the pH, decibel, and Richter scales.

3. With logarithms, you can write an exponential equation in an equivalent logarithmic form.

4. For any positive base b, where b ≠ 1 if and only if x = log y Ex 1. a. Write in logarithmic form. ________________________ b. Write in exponential form. _______________________ c. Write 11 2 = 121 in logarithmic form. _________________________ d. Write log 6 36 = 2 in exponential form. _______________________ e. Write 7 -2 =1/49 in logarithmic form. __________________________ f. Write log 3 1/81= -4 in exponential form. _______________________ 2 = log 11 121 6 2 = 36 Log 7 (1/49) = -2 3 -4 = 1/81

5. You can evaluate logarithms with a base of 10 by using the log key on a calculator. Ex 2. Solve each equation for x. Round your answer to the nearest thousandth. a). 10 x = 1/109 b). x = log 10 1/109 x = -2.037 c). 10 x = 1.498d). 10 x = 7210 x = log 10 1.498 x = log 10 7210 x =.176 x = 3.858

6. The inverse of the exponential function y = 10 x is x = 10 y. To rewrite x = 10 y in terms of y, use the equivalent logarithmic form, y = log 10 x.

7. Examine the tables & graphs below to see the inverse relationship between y=10 x and y = log 10 x. y= 10 x y=x y = log 10 x

8. BeBelow summarizes the relationship between the domain and range of y = 10 x and of y = log 10 X. y = 10 x Domain: all Real #s Range: all positive Real #s y = log 10 X Domain: all positive real #s Range: all Real #s

9. The logarithmic function y = log b x with base b, or x = b y, is the inverse of the exponential function y = b x, where b ≠ 1 and b > 0. One-to-one Property of Exponents If b x = b y, then x = y.

10. Ex. 3 Find the value of v in each equation. B.A.

11. d. v = log 4 64 4 v = 64 4 v = 4 3 (same base) v = 3

12. e. 2 = log v 25 v 2 = 25 v 2 =5 2 v = 5 f. 6 = log 3 v v = 3 6 v = 729 g. v = log 10 1000 10 v = 1000 10 v = 10 3 v = 3 h. 2 = log 7 V V = 7 2 V = 49 I. 1 = log 3 v 3 1 = v 3 = v

16. Homework Pg. 374-375 #12-84 even