## 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**

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