Show that $a^{\log_a n}=n$

Question

Prove that $$a^{\log_a n}=n$$ I read it in a book that it is the 7th law of logarithms. But I don't understand how it actually works.

Answer 1

That's what $\log_ax$ means: it's a number such that $a^{\log_ax}=x$.

Answer 2

It is the definition of logarithm: $x=\log_an$ is indeed that value such that $a^x=n$

Answer 3

People often tend to memorize too many formulas for the logarithms, and get confused.

You need only to focus on two of them, the others can be derived from these:

• $\log_a(x)=\dfrac{\ln(x)}{\ln(a)}$
• $\large{x^y=e^{y\ln(x)}}$

Thus $\quad\Large{a^{\log_a(n)}=e^{\ln(a)\log_a(n)}=e^{\ln(a)\frac{\ln(n)}{\ln(a)}}=e^{\ln(n)}=n}$

Answer 4

taking the logarithm on both sides we get $$\log_a n\ln(a)=\ln(n)$$ since $a\neq 1$ then we have $$\log_a n=\frac{\ln(n)}{\ln(a)}=\log_a n$$

Answer 5

$log_a\ n$ is the number we have to raise $a$ to to get $n$. If we raise $a$ to the number we have to raise $a$ to to get $n$, we get $n$.

Answer 6

Take log with base $a$ both sides. $$(log_an)(log_aa)=log_an$$ $$log_an=log_an$$

Hence $LHS=RHS$ and hence proved