1. Logarithmic Functions Digital Lesson

2. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Definition: Logarithmic Function For x  0 and 0  a  1, y = log a x if and only if x = a y. The function given by f (x) = log a x is called the logarithmic function with base a. 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. Exponential function:y = a x Logarithmic function:y = log a x is equivalent to x = a y A logarithm is an exponent!

3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 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

4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 log 10 –4LOG –4 ENTERERROR no power of 10 gives a negative number Common Logarithmic Function The base 10 logarithm function f (x) = log 10 x is called the common logarithm function. The LOG key on a calculator is used to obtain common logarithms. 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

5. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Properties of Logarithms Examples: Solve for x: log 6 6 = x log 6 6 = 1 property 2  x = 1 Simplify: log 3 3 5 log 3 3 5 = 5 property 3 Simplify: 7 log 7 9 7 log 7 9 = 9 property 3 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 and a log a x = x inverse property

6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 x y Graph f(x) = log 2 x Since the logarithm function is the inverse of the exponential function of the same base, its graph is the reflection of the exponential function in the line y = x. 83 42 21 10 –1 –2 2x2x x y = log 2 x y = x y = 2 x (1, 0) x-intercept horizontal asymptote y = 0 vertical asymptote x = 0

7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Example: f(x) = log 0 x Example: Graph the common logarithm function f(x) = log 10 x. by calculator 10.6020.3010–1–2f(x) = log 10 x 10421x y x 5 –5 f(x) = log 10 x x = 0 vertical asymptote (0, 1) x-intercept

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Graphs of Logarithmic Functions The graphs of logarithmic functions are similar for different values of a. f(x) = log a x (a  1) 3. x-intercept (1, 0) 5. increasing 6. continuous 7. one-to-one 8. reflection of y = a x in y = x 1. domain 2. range 4. vertical asymptote Graph of f (x) = log a x (a  1) x y y = x y = log 2 x y = a x domain range y-axis vertical asymptote x-intercept (1, 0)

9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 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 (x  0, e 2.718281  ) y x 5 –5 y = ln x is equivalent to e y = x

10. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 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 and e ln x = x inverse property 4. If ln x = ln y, then x = y. one-to-one property Examples: Simplify each expression. inverse property property 2 property 1

11. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 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 natural log of both sides inverse property

12. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 Human Memory Model Students participating in a psychology experiment attended several lectures on a subject and were given an exam. Every month for a year after the exam, the students were retested to see how much of the material they remembered. The average scores for the group are given by the human memory model where t is the time in months. (a)What was the average score on the original exam? (b)What was the average score at the end of 2 months? (c)What was the average score at the end of 6 months?

13. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 Homework Page 203-206 25-32 all, 33-37 odd, 43-51 odd, 58-62 even, 63-70 all, 71, 91