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
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