What is the exponent rule of logarithms?

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Answer 1 · 2016-09-16T20:30:46

#log_(a)(m^(n)) = n log_(a)(m)#

Consider the logarithmic number #log_(a)(m) = x#:

#log_(a)(m) = x#

Using the laws of logarithms:

#=> m = a^(x)#

Let's raise both sides of the equation to #n#th power:

#=> m^(n) = (a^(x))^(n)#

Using the laws of exponents:

#=> m^(n) = a^(xn)#

Let's separate #xn# from #a#:

#=> log_(a)(m^(n)) = xn#

Now, we know that #log_(a)(m) = x#.

Let's substitute this in for #x#:

#=> log_(a)(m^(n)) = n log_(a)(m)#