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

Notes

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

• Example 2

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

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

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

• Examples

• 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: $F^{-1}(y)=log_b y$