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

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

\(\sqrt[3]{-125}\)

\(5\)

How to Simplify Fraction Bases

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How to Simplify Decimal Bases

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How to Simplify Advanced Bases

Polynomials

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Complex Numbers

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Matrices

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