3.2 Logarithmic Functions
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Intro Solving for an answer Solving for a baseSolving for an exponent = x2. x 2 = x = 729
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 Now you try… = x 5. x 3 = x = 3125
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!
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 = 4 y y = log = 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
Copyright © by Houghton Mifflin Company, Inc. All rights reserved 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
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 log 10 5 Function ValueKeystrokesDisplay LOG 100 ENTER2 LOG 5 ENTER log 10 ( ) – LOG ( 2 5 ) ENTER
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 Properties of Logarithms 1. log a 1 = 0 since a 0 = 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
Classwork Pg 236 #1-22 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9
Warm-Up Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Graphs of Logarithmic Functions The graph is the inverse of y=a x (reflected over y =x ). VA: x = 0, x-intercept (1,0), increasing 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)
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 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
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 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 ENTER ERRORLN –2 ENTER LN 100 ENTER y = ln x y x 5 –5 y = ln x is equivalent to e y = x
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 Properties of Natural Logarithms 1. ln 1 = 0 since e 0 = 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
Classwork Pg 236 # Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16 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 take the ln of both sides inverse property
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 17 Practice Problems/ Homework Page 236 #23–26, 39-44, 45–59 odd, 79–85 odd