Summary: Logarithmic Functions

Key Equations

Definition of the logarithmic function For [latex]\text{ } x>0,b>0,b\ne 1[/latex], [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] if and only if [latex]\text{ }{b}^{y}=x[/latex].
Definition of the common logarithm For [latex]\text{ }x>0[/latex], [latex]y=\mathrm{log}\left(x\right)[/latex] if and only if [latex]\text{ }{10}^{y}=x[/latex].
Definition of the natural logarithm For [latex]\text{ }x>0[/latex], [latex]y=\mathrm{ln}\left(x\right)[/latex] if and only if [latex]\text{ }{e}^{y}=x[/latex].

Key Concepts

  • The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.
  • Logarithmic equations can be written in an equivalent exponential form using the definition of a logarithm.
  • Exponential equations can be written in an equivalent logarithmic form using the definition of a logarithm.
  • Logarithmic functions with base b can be evaluated mentally using previous knowledge of powers of b.
  • Common logarithms can be evaluated mentally using previous knowledge of powers of 10.
  • When common logarithms cannot be evaluated mentally, a calculator can be used.
  • Natural logarithms can be evaluated using a calculator.

Glossary

common logarithm
the exponent to which 10 must be raised to get x; [latex]{\mathrm{log}}_{10}\left(x\right)[/latex] is written simply as [latex]\mathrm{log}\left(x\right)[/latex]
logarithm
the exponent to which b must be raised to get x; written [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex]
natural logarithm
the exponent to which the number e must be raised to get x; [latex]{\mathrm{log}}_{e}\left(x\right)[/latex] is written as [latex]\mathrm{ln}\left(x\right)[/latex]