Solving the equation involves finding the value of the exponent to which the base 2 must be raised to equal 8. The equation, and others like it, can be solved by finding a logarithm. A logarithm is the exponent to which a base is raised to produce a given value . In general, the exponential form is equivalent to the logarithmic form . So, is equivalent to .
The function , where and , is the parent logarithmic function. The graph of a logarithmic function is a curve that is either strictly increasing or strictly decreasing. The graph also has a vertical asymptote, a vertical line that the curve approaches, but never reaches. Transformations of the parent function can be used to graph other logarithmic functions.
At A Glance
- Finding a logarithm means finding the exponent to which a base must be raised to equal a given value.
- A logarithmic function is the inverse of an exponential function.
- Properties of logarithms can be used to simplify logarithmic expressions.
- To determine the logarithm of a product, add the logarithms of the factors.
- To calculate the logarithm of a quotient, subtract the logarithm of the divisor from the logarithm of the dividend.
- To determine the logarithm of a power, multiply the exponent by the logarithm of the base.
- A logarithm can be changed from one base to another by using the formula .
- The graph of a logarithmic function has a vertical asymptote and is defined on only one side of the asymptote.
- Logarithmic functions can be evaluated to plot points on a graph.
- For a logarithmic function of the form , the value of b determines whether the function is increasing or decreasing.
- Transformations can be used to graph logarithmic functions.