Key Equations

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 $\mathrm{ln}\left(x\right)$.