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# 3.2 Logarithmic Functions

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### 3.2 Logarithmic Functions

1. 1. 3.2 Logarithms • All of the material in this slideshow is important, however the only slides than must be included in your notes are the Recap slides at the end.Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 1
2. 2. Intro Solving for an Solving for a base Solving for an answer exponent 1. 34 = x 2. x2 = 49 3. 3x = 729 81 = x x = ±7 log3729=3 New Calc: Alpha window 5:logBASE( Older Calc: log(729)/log(3) Answer: 6 b/c 36 = 729Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2
3. 3. Now you try… 4. 25 = x 5. x3 = 125 6. 5x = 3125 log 5 3125 = x Plug in calc 32 = x 3 x = 1253 3 log base ans = exp x=5Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3
4. 4. A logarithmic function is the inverse function of an exponential function. Every logarithmic equation has an equivalent exponential form: y = loga x is equivalent to x = a y A logarithm is an exponent!Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4
5. 5. Examples: Write the equivalent exponential equation and solve for y. Logarithmic Equivalent Solution Equation Exponential Equation y = log216 16 = 2y 16 = 24 → y = 4 1 1 1 y = log2( ) = 2y = 2-1→ y = –1 2 2 2 y = log416 16 = 4y 16 = 42 → y = 2 y = log51 1=5y 1 = 50 → y = 0Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5
6. 6. 3.2 Material • Logarithms are used when solving for an exponent. baseexp =ans  logbaseans=exp • In your calculator: log is base 10 ln is base e Properties of Logarithms and Ln 1. loga 1 = 0 2. logaa = 1 3. logaax = x • The graph is the inverse of y=ax (reflected over y =x ) VA: x = 0 , x-intercept (1,0), increasingCopyright © by Houghton Mifflin Company, Inc. All rights reserved. 6
7. 7. The graphs of logarithmic functions are similar for different values of a. f(x) = loga x Graph of f (x) = loga x y-axis y y = ax y=x vertical asymptote y = log2 x x-intercept (1, 0) VA: x = 0 increasing x x-intercept (1, 0) reflection of y = a x in y = xCopyright © by Houghton Mifflin Company, Inc. All rights reserved. 7
8. 8. Example: Graph the common logarithm function f(x) = log10 x. y f(x) = log10 x x 5 (1, 0) x-intercept x=0 vertical –5 asymptoteCopyright © by Houghton Mifflin Company, Inc. All rights reserved. 8
9. 9. y y = ln x The function defined by f(x) = loge x = ln x x 5 is called the natural logarithm function. –5 y = ln x is equivalent to e y = x Use a calculator to evaluate: ln 3, ln –2, ln 100 Function Value Keystrokes Display ln 3 LN 3 ENTER 1.0986122 ln –2 LN –2 ENTER ERROR ln 100 LN 100 ENTER 4.6051701Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9
10. 10. Properties of Natural Logarithms 1. ln 1 = 0 since e0 = 1. 2. ln e = 1 since e1 = e. 3. ln ex = x 4. If ln x = ln y, then x = y. one-to-one property Examples: Simplify each expression. 1 ( ) ln  2  = ln e −2 = − 2 inverse property e  3 ln e = 3(1) = 3 property 2 ln 1 = 0 = 0 property 1Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10
11. 11. −t  1  8223 Example: The formulaR =  12 e (t in years) is used to estimate  10  the age of organic material. The ratio of carbon 14 to carbon 12 1 in a piece of charcoal found at an archaeological dig is R = 15 . 10 How old is it?  1  −t 1  12  e 8223 = 15 original equation 10   −t 10 1 e 8223 = 3 multiply both sides by 1012 −t 10 1 ln e 8223 = ln take the ln of both sides 1000 −t 1 = ln inverse property 8223 1000  1  t = −8223  ln  ≈ −8223 ( − 6.907 ) = 56796  1000  To the nearest thousand years the charcoal is 57,000 years old.Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11
12. 12. 3.2 Material Recap • Logarithms are used when solving for an exponent. logab=c ac =b • In your calculator: log is base 10 ln is base e Properties of Logarithms and Ln 1. loga 1 = 0  because when written in exponential form a0 = 1 2. logaa = 1  because when written in exponential form a1 = a 3. logaax = x because when written in exponential form ax = axCopyright © by Houghton Mifflin Company, Inc. All rights reserved. 12
13. 13. 3.2 Material Recap Graph the function f(x) = log x. y f(x) = log10 x x 5 (1, 0) x-intercept x=0 vertical –5 asymptote • The graph is the inverse of y=ax (reflected over y =x ) VA: x = 0 , x-intercept (1,0), increasingCopyright © by Houghton Mifflin Company, Inc. All rights reserved. 13
14. 14. Practice Problems Page 236 (all work and solutions on blackboard) •#1–16 pick any 4 •17–22 pick any 3 •27–29 all •31–37 pick any 4 •45–52 pick any 3 •53–59 pick any 3 •69–71 all •79–86 pick any 4Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14