Exponent Laws and Logarithm Laws

This purely reference page lists some of the most important exponent laws and logarithm laws, including product law, quotient law, power law, zero law, negative law, one law, reciprocal law, logarithm product law, logarithm quotient law, logarithm power law, logarithm zero law, logarithm negative law, logarithm one law, logarithm reciprocal law.

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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