My textbook seems to be making a big leap when trying to prove the change of base formula for logarithms. If someone could help clear this up it would be very appreciated.
It starts with:
$b^{x \log_b(a)}$
and uses the power rule to get:
$b^{x \log_b(a)} = b^{\log_b (a^x)}$
And it equates all this to:
$b^{x \log_b(a)} = b^{\log_b(a^x)} = a^x$
Okay, I get it up to here, but then for me it leaps from that to this:
$$\log_a(x)\cdot \log_b(a) = \log_b(a^{\log_a(x)}) = \log_b(x)$$
And it says that divide through by $\log_b(a)$ to get the result.
What precisely has happened here? Could someone walk me through this step-by-step?
Thank you.