What Is the Power Property of Logarithms?


Quick Answer

The power property of logarithms states that any exponent in the argument of a logarithmic function can be brought out as a multiplier to the rest of the expression. In algebraic form (note that ? is used to denote a logarithmic base): log?b(x^c) = c * log?b(x) Assuming that log means log with base 10, this can be more simply written as: log(x^c) = c * log(x)

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Full Answer

Spelled out, the power property of logarithms is:

the logarithm of base b of x to the power of c is equivalent to c multiplied by the logarithm of base b of x

Proving this property is true:

First, let: y = log?b(x)

Second, write this in exponent form: x = b^y

Third, raise both sides by the same power: x^a = (b^y)^a

Fourth, revert back into log form: log?b(x^a) = y*a

Finally, substitute back our original replacement of y = log?b(x): log?b(x^a) = a * log?b(x)

The power property of logarithms is one of the four basic properties of logarithms. The other three are: the multiplication property of logarithms, the division property of logarithms and the expansion property of logarithms. Each of these can be proved using methods similar to the one above.

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