Prove the change of base formula for logarithms. That is, prove that log_a (x) = log_b (x) / log_b (a).

Firstly, recall the definition of a logarithm: if y = log_a(x), then this means that y is the power you have to raise a to, to get x, that is a^y = x.Now, we want to introduce a new base, b. Let's take log to base b of both sides of the above equation. We get log_b(a^y) = log_b(x). But remember our rules of logarithms -- we know that ylog_b(a) = log_b(a^y), so we get that ylog_b(a) = log_b(x).Lastly, divide both sides by log_b(a), to obtain: y = log_b(x)/log_b(a). Aha! Remember we started off by saying that y = log_a(x). Therefore, log_a(x) = log_b(x)/log_b(a), and our proof is complete!