I'm currently doing a couple of exercises on logarithmic expressions, and I was a bit confused when presented with the following: $5^{\log_5 17}$.
The answer is $17$, which is the argument of the logarithm in the exponent, but I don't understand the reason why.
A previous question in the exercise was $12^{\log_{12} 144}$, which is pretty straight forward since $\log_{12} 144=2$ and $12^2=144$ but since $\log_5 17$ is an irrational number, I couldn't calculate it the same way.
So my question is, why is $x^{\log_x n}=n$?
[Update]
I've now managed to reason this out and realized how simple it is. Basically, if $y=\log_x n$, then $y$ is the number that $x$ must be raised to to become $n$; therefore, if I raise $x$ to that power $y$, I will naturally get $n$.
I guess my problem was that I tried to work out the question mechanically instead of intelligently and that's why I failed, because once I realized that I can't calculate $\log_5 17$, I was stumped. Now that I reasoned it out, as explained in the previous paragraph, I understand it.