# Key Equations

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

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