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 their 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.
Real-world exponential problems with base 10 can be rewritten as a common logarithm and then evaluated using a calculator.
Natural logarithms can be evaluated using a calculator.

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].