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9.5 – Logarithmic Functions

Key Terms

  • Common Logarithm – A logarithm with base 10.
    • The common logarithm of x is the power to which 10 must be raised to equal x.
    • It is written as log_{10} x, or often simply log x.
  • Exponential Equation – An equation that contains an exponential expression.
  • Inverse – A function or relation that undoes the work of another function or relation.
    • The input of a relation is the same as the output of its inverse.
  • Logarithm – The inverse of an exponential expression. y=b^{x} is the inverse of x=log_b y, which is the logarithm, base b, of y.
  • Logarithmic Equation – An equation that contains a logarithm.
  • Logarithmic Function – The inverse of an exponential function.
    • In general, the inverse of F(x)=b^{x} is F^{-1}(y)=log_b y, which is the logarithm, base b, of y.
  • Natural Logarithm – The logarithm with base e, where e\approx 2.718.
    • The natural logarithm of x is usually written as ln x.
  • Vertical Line Test – A method for testing whether a given graph is a function.
    • If a vertical line can be drawn that intersects the graph in two or more points, then the graph is not a function.
    • If no such line can be drawn, the graph is a function.

Review

Exponential Functions
  • Exponential functions are written in the form F(x)=b^{x}.
  • The vertical line test shows that the inverse of an exponential function (a logarithmic function) is also a function.
  • An exponent of F^{-1} is different than the inverse of a function F^{-1}.
    • Exponents: F^{-1}=\frac{1}{F}
    • Functions: F^{-1}=\; Inverse\; of\; F

  • Graph of an Exponential Function

Alg2B 9.5 Exp Fx Graph

Notes

Logarithmic Functions
  • A logarithmic function is the inverse of an exponential function.
    • It undoes an exponential function.
  •  Logarithmic Function Graph

Alg2B 9.5 Log Fx Graph

Graphs of Exponential and Logarithmic Functions
  • The graphs of these functions are flipped over the line y = x.
  • The x- and y-coordinates of each point switches.
  • The output of the functions become the input of the inverse functions.
  • Example 1

Alg2B 9.5 Graph Fx Ex1

  • Example 2

Alg2B 9.5 Graph Fx Ex2

 

The Relationship Between Logarithms and Exponents
  • Exponential and logarithmic equations are all about multiplication. In both cases:
    • b = the number to multiply
    • c = the number of times to multiply
    • a = the product
Alg2B 9.5 Finding Logs and Exps

 

Logarithms
  • log_3 81=4 can be said many different ways:
    • “The logarithm of 81 with base 3 is 4.”
    • “Log base 3 of 81 is 4.”
    • “The base 3 log of 81 is 4.”
  • The logarithmic equation log_b a=c can be rewritten as the exponential equation b^{c}=a.
    • b became the base of an exponent
    • c became the exponent
    • a ended up on the other side of the equation by itself

Alg2B 9.5 Log Examples

 

Logarithms and Natural Logs: Two Special Bases
  • Two special bases for logarithms are 10 and the number e.

Alg2B 9.5 Log Ln

Alg2B 9.5 Log Review

  • Examples

Alg2B 9.5 Special Bases

  • Base 10: Our number system is based on powers of 10.
    • Common Logarithm: Log base 10 is written as log x instead of log_{10} x.
    • Whenever you see a logarithm with no base, it is the common base 10 logarithm.
  • Base e: This is often used as the base of an exponential function.
    • Natural Logarithm: Log base e is written as ln x. (lowercase L).
    • e ≈ 2.718.
    • So the natural logarithm is approximately log_{2.718}.

Examples

  • Ex 1. Which logarithmic equation is equivalent to the exponential equation below?
    • 4^{c}=64
      • Answer: log_4 64=c
  • Ex 2. Which logarithmic equation is equivalent to the exponential equation below?
    • e^{a}=28.37
      • Answer: ln 28.37 = a
  • Ex 3. Which exponential equation is equivalent to the logarithmic equation below?
    • c = ln 4
      • Answer: e^{c}=4
  • Ex 4. Which logarithmic equation is equivalent to the exponential equation below?
    • 67.21=e^{a}
      • Answer: ln 67.21 = a
  • Ex 5. Which exponential equation is equivalent to the logarithmic equation below?
    • log 987 = a
      • Answer: 10^{a}=987
  • Ex 6. It is true that the exponential function is the inverse of a logarithmic function.
  • Ex 7. Which function is the inverse of F(x)=b^{x}?
    • Answer: F^{-1}(y)=log_b y

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