Logarithmic Expressions (College Board AP® Precalculus): Study Guide

Definition of a logarithm

What is a logarithm?

• A logarithm answers the question: "What power must I raise a base to, in order to get a particular value?"

• In the expression ${\log }_{b}c$

  • $b$ is the base

    • and $c$ is the 'particular value'

  • So the value of that expression represents

    • the exponent (power) that the base $b$ must be raised to

    • in order to produce the value $c$

  • The base $b$ must satisfy $\mathit{b}\mathbf{>}\mathbf{0}$ and $\mathit{b}\mathbf{\ne }\mathbf{1}$

  • The value $c$ must be positive

    • you cannot take the logarithm of zero or a negative number

• In other words

  • ${\mathbf{log}}_{\mathit{b}}\mathit{c}\mathbf{=}\mathit{a}$

    • means exactly the same thing as ${\mathit{b}}^{\mathit{a}}\mathbf{=}\mathit{c}$

  • These two forms (logarithmic form and exponential form) are just two equivalent ways of writing the same relationship

• Here are some examples:

What is the common logarithm?

• When no base is written, the logarithm is understood to have base 10

  • This is called the common logarithm

• E.g. $\log 100$ means the same thing as ${\log }_{10}100$

  • and it is equal to 2 because ${10}^2=100$

• The common logarithm appears frequently on calculators and in applications

How do you evaluate logarithmic expressions?

• Some logarithmic values can be worked out using basic arithmetic

  • i.e. by thinking about what power of the base gives the desired value

  • E.g. ${\log }_{2}32$

    • since $2^5=32$, it follows that ${\log }_{2}32=5$

  • E.g. ${\log }_4\frac{1}{16}$

    • since $4^{-2}=\frac{1}{4^2}=\frac{1}{16}$, it follows that ${\log }_4\frac{1}{16}=-2$

  • E.g. ${\log }_{9}3$

    • since $9^{\frac{1}{2}}=\sqrt{9}=3$, it follows that ${\log }_{9}3=\frac{1}{2}$

• Other values cannot be evaluated exactly by hand

  • but can be estimated using technology (a calculator)

  • E.g. ${\log }_{2}5$

    • There is no integer or simple fraction $a$ such that $2^a=5$ exactly

    • But a calculator will give you the value

      • ${\log }_{2}5=2.321928...$

Logarithmic scales

What is a logarithmic scale?

• On a standard (linear) scale, each unit represents the same additive change

  • E.g. the marks might be at $0,1,2,3,4,\dots$

    • with equal spacing between them

• On a logarithmic scale, each unit represents the same multiplicative change

  • specifically, a multiplication by the base of the logarithm

• E.g. on a base-10 logarithmic scale

  • the marks might be at

    • ${10}^0=1$, ${10}^1=10$, ${10}^2=100$, ${10}^3=1000$, $\dots$

  • Each step along the scale multiplies the value by 10, rather than adding a fixed amount

• Logarithmic scales are useful for displaying data that would otherwise span a very wide range of values

• Examples of commonly-used logarithmic scales include:

  • earthquake magnitudes (the Richter scale)

  • sound intensity (decibels)

  • and the pH scale in chemistry

How does a logarithmic scale work?

• On a logarithmic scale with base $b$

  • the position of a value on the scale is its logarithm base $b$

• E.g. on a base-10 scale

  • the value 1 sits at position 0 (since ${\log }_{10}1=0$)

  • the value 10 sits at position 1

  • the value 100 sits at position 2

  • etc.

• Equal spacing on the scale corresponds to equal ratios in the actual values

  • not equal differences

• E.g. the distance from 10 to 100 on a base-10 log scale is the same as the distance from 100 to 1000

  • both represent a multiplication by 10

• Values that are close together on a linear scale (like 0.01 and 0.001) may be far apart on a logarithmic scale

  • and values that seem vastly different on a linear scale (like 1000 and 10,000) may be close together on a logarithmic scale