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)
Continue ReadingSpelled 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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