The logarithm of a number is the exponent to which a base must be raised to produce that number.
In other words, the logarithm is the inverse operation of exponentiation.
Logarithmic Notation
The logarithm of a number \$a\$ with base \$b\$ is denoted as \$\log_b a\$.
- The expression \$\log_b a = c\$ means that \$b^c = a\$.
- Here, \$b\$ is the base, \$a\$ is the result, and \$c\$ is the exponent.
If the base is not specified, it is assumed to be 10. This is called the common logarithm.
Converting Between Logarithmic and Exponential Forms
Logarithmic and exponential forms are two ways of expressing the same relationship.
- The logarithmic form \$\log_b a = c\$ is equivalent to the exponential form \$b^c = a\$.
- Converting between these forms is a key skill in solving logarithmic and exponential equations.
To convert from logarithmic to exponential form, remember that the base of the logarithm becomes the base of the exponent.
Calculating Logarithms with a Calculator
Most calculators have a log button for base 10 logarithms and an ln button for natural logarithms (base \$e\$).
- For other bases, use the change of base formula: \$\log_b a = \frac{\log a}{\log b}\$ or \$\log_b a = \frac{\ln a}{\ln b}\$.
The number \$e\$ is an important mathematical constant approximately equal to 2.71828. It is the base of natural logarithms.
Solving Exponential Equations Using Logarithms
Logarithms can be used to solve exponential equations by converting them into logarithmic form.
TipWhen solving exponential equations, first isolate the exponential term before applying logarithms.
Solving Multistep Exponential Equations
Some exponential equations require additional algebraic steps to isolate the exponential term.
NoteIf the exponent is a more complex expression, additional algebraic steps may be needed after applying logarithms.
Leaving Solutions in Unsimplified Form
In some cases, solutions to exponential equations are left in logarithmic form, especially in multiple-choice questions.
NoteThe solution can also be verified by calculating the numerical value and checking against the answer choices.
Graphing Logarithmic Functions
The graph of a logarithmic function \$y = \log_b x\$ is the reflection of the exponential function \$y = b^x\$ over the line \$y = x\$.
NoteThe domain of a logarithmic function \$y = \log_b x\$ is \$x > 0\$, and it has a vertical asymptote at \$x = 0\$.
Self review1. Evaluate \$\log_3 81\$ and convert it to exponential form. 2. Solve the exponential equation \$5^x = 125\$ using logarithms. 3. Graph the functions \$y = \log_2 x\$ and \$y = 2^x\$ on the same axes. Describe their relationship.
Theory of KnowledgeHow do logarithms, as the inverse of exponentiation, illustrate the concept of mathematical inverses? What other mathematical operations have inverses, and how are they used to solve equations?
