Logarithm property needs an explanation

Question

So here's the property: (Excuse the spacing, I'm just trying to make it clearer)

$\log_a{x} = \log_{10} {x} / \log_{10} {a}$

What I don't understand is WHY it's correct, can someone please give me some proof/explanation? Thank you so much for your help.

Answer 1

Consider this rule: $\log_a(b)\cdot c = \log_a(b^c)$. Replace $c$ by $\log_b(c)$, and you get

$$\log_a(b) \cdot \log_b(c) = \log_a(b^{\log_b(c)}) = \log_a(c).$$

Answer 2

One may notice that $$ \log_a(x)=\frac{\log x}{\log a}=\frac{\frac{\log x}{\log 10}}{\frac{\log a}{\log 10}}=\frac{\log_{10} x}{\log_{10} a}. $$

Answer 3

Here’s an explanation that’s perhaps a bit more consistent with the way that the logarithms are first introduced in school:

$N=\log_ax$ means $a^N=x$.
$M=\log_{10}a$ means $10^M=a$.
Thus $x=\left(10^M\right)^N=10^{MN}$.
Consequently, $\log_{10}x=MN=\log_{10}a\cdot\log_ax$.

And there you are!