# 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.

## 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].