Exponent Laws and Logarithm Laws

This purely reference page lists some of the most important exponent laws and logarithm laws, and a quick review of logarithms, including natural log, $l n 𝑥$ .

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Exponent Laws

𝑥 0 = 1 1 𝑥 𝑛 = 𝑥 - 𝑛 𝑥 𝑛 \cdot 𝑥 𝑚 = 𝑥 𝑛 + 𝑚 𝑥 𝑛 𝑥 𝑚 = 𝑥 𝑛 - 𝑚 (𝑥 𝑛) 𝑚 = 𝑥 𝑛 \cdot 𝑚 (𝑥 𝑦) 𝑛 = 𝑥 𝑛 𝑦 𝑛 𝑛 \sqrt 𝑥 = 𝑥 1 / 𝑛 𝑛 \sqrt 𝑥 𝑚 = 𝑥 𝑚 / 𝑛 𝑛 \sqrt 𝑥 𝑦 = 𝑛 \sqrt 𝑥 𝑛 \sqrt 𝑦

[Need to know how to differentiate $𝑥 𝑛$ ? Visit our free Derivative of Exponential Functions page!]

Exponential and Log Functions

Quick review: What is a logarithm?

A log function "undoes" an exponential function.

For example, since $23 = 8$ , we have $l o g 2 8 = 3 .$

We express this idea mathematically as

𝑎 𝑦 = 𝑥 ⟺ l o g 𝑎 𝑥 = 𝑦

Because of this "undoing," we know:

l o g 𝑎 𝑎 𝑥 = 𝑥 a n d 𝑎 l o g 𝑎 𝑥 = 𝑥

Natural log, $l n 𝑥$

The log with base $𝑒,$ where $𝑒 = 2.71828 \dots$ is known as the natural log, $l n 𝑥 .$
That is,

l n 𝑥 = l o g 𝑒 𝑥

and $l n 𝑒 = 1 .$

Hence

l n 𝑒 𝑥 = 𝑥 a n d 𝑒 l n 𝑥 = 𝑥

Logarithm Laws

l o g 𝑎 𝑥 𝑦 = l o g 𝑎 𝑥 + l o g 𝑎 𝑦 l o g 𝑎 𝑥 𝑦 = l o g 𝑎 𝑥 - l o g 𝑎 𝑦 l o g 𝑎 𝑥 𝑛 = 𝑛 l o g 𝑎 𝑥

Change Logarithm Base

l o g 𝑎 𝑥 = l o g 𝑏 𝑥 l o g 𝑏 𝑎