[Math] Simplifying two logarithms with different bases

Question

I am being asked to simplify:

$(\log_4 7)(\log_7 5)$

How can this be simplified given that the bases are different?

Answer 1

Here's a way that may be the easiest to understand, using the change-of-base formula in its simplest form: $$ (\log_4 7)(\log_7 5) = \frac{\log_e 7}{\log_e 4} \cdot \frac{\log_e 5}{\log_e 7} = \frac{\log_e 5}{\log_e 4} = \log_4 5. $$

Here's a way that uses a corollary of the change-of-base formula: $$ \underbrace{(\log_4 7)(\log_7 5) = (\log_7 7)(\log_4 5)}_\text{a corollary of the change-of-base formula} = 1\cdot\log_4 5. $$

Here's the corollary: $$ (\log_A P)(\log_B Q)(\log_C R)(\log_D S)\cdots = (\log_D P)(\log_A Q)(\log_B R)(\log_C S)\cdots $$ and generally you can permute the subscripts $A,B,C,D,\ldots$ in any way at all while leaving the arguments $P,Q,R,S,\ldots$ where they are, without changing the value of the product.

That follows from the change-of-base-formula, which is actually a special case of it.