# What is a Base?

## Base Definition

In a power, the base is the large number underneath the exponent. When you simplify the power, you will this number by itself multiple times (as determined by the exponent).

For example, $$3^{2}$$ means that you should multiply 3 by itself two times.

$$3^{2}=3\times3=9$$

## Base of a Logarithm

In a logarithm, the base is the small number written in a subscript immediately after the log function.

Logarithms allow us to find the missing exponent in a power. They can be written in exponential form as explained on this page. The exponential form of the logarithm above is...

## Logarithmic Form

$$log_3{81}=x$$

## Exponential Form

$$3^{x}=81$$

The answer to this logarithm would be 4 because $$3^{4}=81$$. The base of the logarithm is the same as the base of the power when the logarithm is written in exponential form.

## How to Simplify Negative Bases

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When you have a negative base with an exponent, you simplify the power

### Positive Exponent with a Negative Base

$$(-4)^{3}$$

Because we have a positive 3 in the exponent, we will multiply -4 by itself 3 times.

$$(-4)^{3}= (-4)(-4)(-4)$$

Three negatives multiplied together is a negative.

$$-64$$

### Negative Exponent with a Negative Base

$$(-5)^{-2}$$

$$\frac{1}{5^{2}})$$

$$\frac{1}{5\times5})$$

$$\frac{1}{25})$$

### Fraction Exponent with a Negative Base

$$(-125)^{\frac{1}{3}}$$

$$\sqrt[3]{-125}$$

$$5$$

Polynomials

Complex Numbers

Matrices