Proving Logarithims

Question

I'm trying to figure out how to go about proving this statement:

$$n^{\log(a)} = a^{\log(n)}$$

I'm told that I cannot prove from both sides. I tried to $\log$ the first side to get:

$\log(n)\log(a)$

But I'm not sure where to go from here, any ideas would be much appreciated.

Answer 1

We can just do some simple manipulation to get the result. We can take a log and exponentiate the left side:

$$n^{\log(a)}=e^{\log(n^{\log(a)})}=e^{\log(a)\log(n)}=e^{\log(n)\log(a)}=e^{\log(a^{\log(n)})}=a^{\log(n)}.$$

Answer 2

I'm not sure if you can use it, but it is well known that

$$x^y = e^{(\ln x)y} $$

where $\ln$ is the natural logarithm and $e$ is Eulers constant. Just apply this.

Answer 3

Suppose $n=a^x$, then $\log n=x\log a$ by taking logs.

Then, by substitution, $n^{\log a}=(a^x)^{\log a}=\dots$

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How to find the method - well you start with $n$ to some power, and you want to end with $a$ to some power, so it is natural to express $n$ as a power of $a$, and once this is done the result drops out.