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Exponent Laws and Logarithm Laws

Handy Exponent and Logarithm Rules

Below is a handy summary of exponent rules and logarithm rules.


Exponent Laws

\begin{align*}
x^0 &= 1 \\[4px]
\dfrac{1}{x^n} &= x^{-n} \\[8px]
x^n \cdot x^m &= x^{n+m} \\[8px]
\dfrac{x^n}{x^m} &= x^{n-m} \\[8px]
\left(x^n \right)^m &= x^{n \cdot m} \\[8px]
\left(xy\right)^n &= x^n y^n \\[8px]
\sqrt[n]{x} &= x^{1/n} \\[8px]
\sqrt[n]{x^m} &= x^{m/n} \\[8px]
\sqrt[n]{\dfrac{x}{y}} &= \dfrac{\sqrt[n]{x}}{\sqrt[n]{y}}
\end{align*}
Need to know how to differentiate $x^n$? Visit our free Calculating Derivatives: Problems & Solutions page!


Exponential and Log Functions

• Quick review: What is a logarithm?
A log function “undoes” an exponential function.

For example, since $2^3 = 8$, we have $\log_2 8 = 3.$

We express this idea mathematically as

\[a^y = x \iff \log_a x = y \]
Because of this “undoing,” we know:
\[ \log_a a^x = x \quad \text{and} \quad a^{\log_a x} = x \]

• Natural log, ln(x)
The log with base $e,$ where $e = 2.71828\ldots$ is known as the natural log, $\ln x.$

That is,
\[\ln x = \log_e x \]
and $\ln e =1.$

Hence
\[ \ln e^x = x \quad \text{and} \quad e^{\ln x} = x \]


Logarithm Laws


\begin{align*}
\log_a xy &= \log_a x + \log_a y \\[8px]
\log_a \dfrac{x}{y} &= \log_a x\, – \log_a y \\[8px]
\log_a x^n &= n \log_a x
\end{align*}

• Change Logarithm Base

\[\log_a x = \dfrac{\log_b x}{\log_b a}\]