Logarithmic Functions



Solving the equation 2x=82^x = 8 involves finding the value of the exponent xx 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 xx to which a base bb is raised to produce a given value yy. In general, the exponential form y=bxy=b^x is equivalent to the logarithmic form x=logbyx=\log_{b}y. So, 23=82^{3} = 8 is equivalent to log28=3\log_{2}8=3.

The function f(x)=logbxf(x)=\log_b{x}, where b>0b>0 and b1b\neq1, 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 f(x)=logbxf(x)=\log_b{x} 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 logbx=logaxlogab\log_b{x}=\frac{\log_a{x}}{\log_a{b}}.
  • 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 f(x)=logbxf(x)=\log_{b}x, the value of b determines whether the function is increasing or decreasing.
  • Transformations can be used to graph logarithmic functions.