Why is log(x^n) = n log(x)?

[nobb]

Hey
I am doing an investigation for logarithms, and I have a question. logx^n = nlogx. Based on previous knowledge of exponents, could someone please explain why this is true? Thanks.

 

[recon]

Let lgx = m. Then, 10^m = x.

x^r = (10^m)^r
x^r = 10^rm
lgx^r = rm
lgx^r = rlgx

 

[nobb]

I don't really get what you are doing. You are solving it with knowledge of logs. Is it possible to answer the question with knowledge from exponents only? Or could you please tell me how this relates to exponents?

 

[Galileo]

Use the identity:
$$a^x=y \iff x=\log_ay$$
Then with the familiar rules for manipulating exponentials, the rules for logarithms follow automatically.

 

[recon]

OK, I try again.

Let lgx = m. Then,

10^m = x. <--- This is taken directly from the definition of logs.

x = 10^m

x^r = (10^m)^r <-- follows from the rule that says if a = b, then a^r = b ^r

x^r = 10^rm <-- follows from the rule that says (a^r)^m = a^rm

lgx^r = rm <--- from the definition of logs again.

lgx^r = rlgx <--- Remember the definition we gave on the first line of this post that states that m = lgx?

 

[HallsofIvy]

There are a number of DIFFERENT ways to prove log(xⁿ)= n log(x)
depending on exactly how you are DEFINING "log(x)". What is your definition of
log(x)??