1. 3.2 Logarithmic Functions

2. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Intro Solving for an answer Solving for a baseSolving for an exponent 1. 3 4 = x2. x 2 = 493. 3 x = 729

3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 Now you try… 4. 2 5 = x 5. x 3 = 125 6. 5 x = 3125

4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Definition: Logarithmic Function Every logarithmic equation has an equivalent exponential form: y = log a x is equivalent to x = a y A logarithmic function is the inverse function of an exponential function. A logarithm is an exponent!

5. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Examples: Write Equivalent Equations y = log 2 ( ) = 2 y Examples: Write the equivalent exponential equation and solve for y. 1 = 5 y y = log 5 1 16 = 4 y y = log 4 16 16 = 2 y y = log 2 16 SolutionEquivalent Exponential Equation Logarithmic Equation 16 = 2 4  y = 4 = 2 -1  y = –1 16 = 4 2  y = 2 1 = 5 0  y = 0

6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 3.2 Material Logarithms are used when solving for an exponent. base exp =ans  log base ans=exp In your calculator: log is base 10 ln is base e Properties of Logarithms and Ln 1.log a 1 = 0 2.log a a = 1 3.log a a x = x The graph is the inverse of y=a x (reflected over y =x ) VA: x = 0, x-intercept (1,0), increasing

7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 log 10 –4LOG –4 ENTERERROR no power of 10 gives a negative number Common Logarithmic Function Examples: Calculate the values using a calculator. log 10 100 log 10 5 Function ValueKeystrokesDisplay LOG 100 ENTER2 LOG 5 ENTER0.6989700 log 10 ( ) – 0.3979400LOG ( 2 5 ) ENTER

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Properties of Logarithms Examples: 1. Solve for x: log 6 6 = x 2. Simplify: log 3 3 5 Properties of Logarithms 1. log a 1 = 0 since a 0 = 1. 2. log a a = 1 since a 1 = a. 4. If log a x = log a y, then x = y. one-to-one property 3. log a a x = x

9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Graphs of Logarithmic Functions The graphs of logarithmic functions are similar for different values of a. f(x) = log a x x-intercept (1, 0) increasing reflection of y = a x in y = x VA: x = 0 Graph of f (x) = log a x x y y = x y = log 2 x y = a x y-axis vertical asymptote x-intercept (1, 0)

10. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 Example: f(x) = log 0 x Example: Graph the common logarithm function f(x) = log 10 x. y x 5 –5 f(x) = log 10 x x = 0 vertical asymptote (1, 0) x-intercept

11. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Natural Logarithmi c Function The function defined by f(x) = log e x = ln x is called the natural logarithm function. Use a calculator to evaluate: ln 3, ln –2, ln 100 ln 3 ln –2 ln 100 Function ValueKeystrokesDisplay LN 3 ENTER1.0986122 ERRORLN –2 ENTER LN 100 ENTER4.6051701 y = ln x y x 5 –5 y = ln x is equivalent to e y = x

12. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 Properties of Natural Logarithms 1. ln 1 = 0 since e 0 = 1. 2. ln e = 1 since e 1 = e. 3. ln e x = x 4. If ln x = ln y, then x = y. one-to-one property Examples: Simplify each expression. inverse property property 2 property 1

13. Classwork Pg 236 # 1-21 odd Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13

14. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 Example: Carbon Dating Example: The formula (t in years) is used to estimate the age of organic material. The ratio of carbon 14 to carbon 12 in a piece of charcoal found at an archaeological dig is. How old is it? To the nearest thousand years the charcoal is 57,000 years old. original equation multiply both sides by 10 12 take the ln of both sides inverse property

15. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15 3.2 Material Logarithms are used when solving for an exponent. log a b=c  a c =b In your calculator: log is base 10 ln is base e Properties of Logarithms and Ln 1.log a 1 = 0 2.log a a = 1 3.log a a x = x The graph is the inverse of y=a x (reflected over y =x ) VA: x = 0, x-intercept (1,0), increasing

16. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16 Practice Problems Page 236 #2-22 even, 23–29, 39-44, 45–59 odd, 79–85 odd