1. 7.4: Evaluate Logarithms and Graph Logarithmic Functions Objectives: Write equivalent forms for logarithmic and exponential functions Write, evaluate, and graph logarithmic functions Common Core Standards: F-IF-5, F-IF-7, F-BF-3, F-BF-4, F-BF-5, N-RN-2 Assessments: Define all vocab for this section Do worksheet 7-4

2. How many times would you have to double $1 before you had $8? You could use an exponential equation to model this situation. 1(2 x ) = 8. You may be able to solve this equation by using mental math if you know 2 3 = 8. So you would have to double the dollar 3 times to have $8.

3. How many times would you have to double $1 before you had $512? You could solve this problem if you could solve 2 x = 8 by using an inverse operation that undoes raising a base to an exponent equation to model this situation. This operation is called finding the logarithm. A logarithm is the exponent to which a specified base is raised to obtain a given value.

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

5. 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 Exponential Equation Logarithmic Form 3 5 = 243 25 = 5 10 4 = 10,000 6 –1 = a b = c 1 6 1 2 log 3 243 = 5 1 2 log 25 5 = log 10 10,000 = 4 1 6 log 6 = –1 log a c =b

6. A logarithm is an exponent, so the rules for exponents also apply to logarithms. You may have noticed the following properties in the last example.

7. Log b When there is no base written in the log equation, the base is 10. – Log 10 b, but you don’t need to write the 10

8. Most calculators calculate logarithms only in base 10 or base e (see Lesson 7-6). You can change a logarithm in one base to a logarithm in another base with the following formula.

9. 4 log a.64 Evaluate the logarithm.

10. Natural Logarithm (ln) Log e x = ln x The calculator has this button on the left side.

11. Solve Log 2 32 Log 27 3 Log 12 ln 0.75

12. Solve Log 8Log 4

13. EXAMPLE 4 Tornadoes The wind speed s (in miles per hour) near the center of a tornado can be modeled by: where d is the distance (in miles) that the tornado travels. In 1925, a tornado traveled 220 miles through three states. Estimate the wind speed near the tornado’s center. 93 log d + 65 s =

14. Solve a (2x+1) = a (x-4) When you have an equation with the same base on both sides of the =, you can set the exponents equal to each other and solve. 2x+1 = x-4 x= -5 You can do the opposite as well, and make both sides an exponent of the same base – e x = e -5

15. Exponential and logarithmic operations undo each other since they are inverse operations.

16. e and ln are inverse functions of each other. Log b x and b x are inverses and will cancel each other out – Log b b x = x – Log 3 3 2 = 2 ln will cancel out e and vice versa – Ln*e 2 = 2(log e e 2 = 2) – e ln2 = 2 To use the inverse to solve for y. – x= ln(y+1) – e x = e ln(y+1) Make both the exponent of e – e x = y+1e ln cancel leaving just (y+1) – e x -1 = ysubtract 1

17. Find the Inverse of the function y= ln (x+3) – x= ln (y+3) Switch x and y – e x = y+3Write in exponential form – e x -3 = ySubtract 3 – y= e x -3 is the inverse of the function

18. Inverse

19. 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>0}. The domain of y=log 2 x is {x > 0}, and the range is all real numbers ( R ).

20. Graphing Logarithmic Functions Log b x is increasing if b>1 and decreasing 0<b<1. Because it is the inverse of b x, it will reflect across the line y=x. Graph Log (x) Graph Log.5 (x) (how do we enter this into the calculator???)