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Logarithmic Functions

Logarithmic Functions. Think about it…. Is there an inverse of f(x)=a x The function is 1-1 (and passes the horizontal line test) then the function has an inverse The inverse is called the logarithmic function f -1 The base would be a. Definition. a is a positive number where a≠1

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Logarithmic Functions

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  1. Logarithmic Functions

  2. Think about it… • Is there an inverse of f(x)=ax • The function is 1-1 (and passes the horizontal line test) then the function has an inverse • The inverse is called the logarithmic function f-1 • The base would be a

  3. Definition • a is a positive number where a≠1 • The logarithmic function with base a is denoted by: • And is defined by

  4. Forms • We can use this definition to switch between exponential form and logarithmic form exponent exponent base base

  5. Evaluating Logarithms • Remember to switch between forms • If one form is true then so is the other • Evaluate log10100,000=5 b/c 105=100,000 log28=3 b/c 23=8 log2(1/8)=-3 b/c 2-3=1/8

  6. Graphing Logarithmic Functions • Make a table of values • Ex: f(x)=log2x remember….2y=x

  7. Translations of logarithms • Same rules apply!! • Vertical • Horizontal • Reflection over the x axis • Reflection over the y axis

  8. Properties of Logarithms • loga1=0 • logaa=1 • Logaax=x • alogax=x

  9. Common Logs • A log with base of 10 is called the common log • logx=log10x • Follow the same properties

  10. Natural Logs • Logarithm with base e • ln x=logex • Properties: • ln 1=0 • lne=1 • ln ex=x • elnx=x

  11. Graphing Natural Logs f(x)=ln(x) • Make a table of values

  12. Translating natural log function • Vertical Translations • ln(x)±c • Horizontal Translations • ln(x±c) • Vertical Stretch/Compression • cln(x) • Horizontal Stretch/Compression • ln(cx) • Reflection over the x-axis • -ln(x) • Reflection over the y-axis • ln(-x) • Reflection over y=x • x=ln(y) ex=y

  13. Practice/Hw • Practice • 8-3: p.108 # 1, 5, 13, 21, 30, 42, 50 • Hw • P. 405 #1-28

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