1. 3.2 Logarithmic Functions and Their Graphs We know that if a function passes the horizontal line test, then the inverse of the function is also a function. So, an exponential function f(x) = b x, has an inverse that is a function. This inverse is the logarithmic function with base b, denoted or Read as “log base b of x”

2. Changing Between Logarithmic and Exponential Form If x > 0 and 0 < b ≠ 1, then if and only if Read as “y equals log base b of x” Which means that y is the exponent to which b must be raised to obtain x

3. Changing Between Logarithmic and Exponential Forms Exponential FormLogarithmic Form y is the exponent b is the base x is the result of raising b to the y power

4. Evaluating Logarithms a.) because b.) because c.) because Rewrite radicals as a rational exponential expressions Rewrite fractions as a rational exponential expressions

5. Evaluating Logarithms d.) because e.) because

6. Basic Properties of Logarithms For 0 0, and any real number y, log b 1 = 0 because b 0 = 1 log b b = 1 because b 1 = b log b b y = y because b y = b y because

7. Evaluating Logarithmic and Exponential Expressions a) b) c)

8. Common Logarithms – Base 10

9. Basic Properties of Common Logarithms

10. Evaluating Logarithmic and Exponential Expressions with Base 10

11. Evaluating Logarithms with a Calculator Most calculators have a key for common logs and natural logs. The TI-84 is capable of performing operations on bases other than 10 or e. Generally, round to the thousandths place.

12. Solving Simple Logarithmic Equations

13. Natural Logarithms – Base e

14. Basic Properties of Natural Logarithms Let x and y be real numbers with x > 0 ln 1 = 0 because e 0 = 1 ln e = 1 because e 1 = e ln e y = y because ln and e are inverses e ln x = x because ln and e are inverses

15. Evaluating Logarithmic and Exponential Expressions with Base e

16. Graphs of Logarithmic Functions and Natural Logarithmic Functions Since logs and exponentials are inverses the domain and range switch!…the x values and y values are exchanged… They are also symmetric about the line y = x

19. f xy = 2 x –3 1 8 –2 1 4 –1 1 2 01 12 24 38 f x = 2y 1 8 –3 1 4 –2 1 2 –1 10 21 42 83 Orderedpairs reversed y x y 510–5 5 10 –5 f -1 x = 2 y or y = log 2 x f y = 2 x y =x DOMAIN of = (– ,  ) = RANGE of RANGE of f = (0,  ) = DOMAIN of Logarithmic Function with Base 2 f f -1

20. Transforming Logarithmic Graphs Describe how to transform the graph of y = ln x or y = log x into the graph of the given function. a) g(x) = ln (x + 2) The graph is obtained by translating the graph of y = ln (x) two units to the LEFT.

21. Transforming Logarithmic Graphs Describe how to transform the graph of y = ln x or y = log x into the graph of the given function. b) h(x) = ln (3 - x) The graph is obtained by applying, in order, a reflection across the y-axis followed by a transformation three units to the RIGHT.

22. Transforming Logarithmic Graphs Describe how to transform the graph of y = ln x or y = log x into the graph of the given function. c) g(x) = 3 log x The graph is obtained by vertically stretching the graph of f(x) = log x by a factor of 3.

23. Transforming Logarithmic Graphs Describe how to transform the graph of y = ln x or y = log x into the graph of the given function. d) h(x) = 1+ log x The graph is obtained by a translation 1 unit up.

24. Practice!!!!! Section 3.2 Logarithmic Functions and Their Graphs Pg 236 Vocabulary Check 1-5 Pg 236 Exercises 1-25 every other odd, 27-31, 35, 39-44, 45-63 every other odd, 65-73 odds, 77-85 odds, 88, 89 Try a problem from each section and don’t forget some applications!