What log rule was used to simply this expression?

Question

I'm unclear how the left side is equal to the right side.

$$365\log(365) - 365 - 305\log(305) + 305 - 60\log(365) = 305\log\left(\frac{365}{305}\right)-60$$

I know $\log(a) - \log(b) = \log (a/b)$ but if you stick constants before each ln() then how do you apply the rule to get 305 as the constant on the right side of the equation?

Answer 1

There are a couple of steps missing.

\begin{align*} &\quad\ 365\log(365) - 365 - 305\log(305)+305-60\log(365)\\ &= [365\log(365) -60\log(365)] + [-365 + 305] - 305\log(305)\\ &= 305\log(365) - 60 - 305\log(305)\\ &= [305\log(365) - 305\log(305)] - 60\\ &= 305[\log(365) - \log(305)] - 60\\ &= 305\log(365/305) - 60 \end{align*}

Answer 2

Collect the constants (-365 + 305 = -60), and the terms with $log(365)$: $$ 365\log(365) - 365 - 305\log(305) + 305 - 60\log(365) = 305\log(365) - 305\log(305) - 60 $$

Now factor out 305, and use the identity you mentioned: $$ 305(\log(365) - \log(305)) - 60 = 305\log(365/305) - 60 $$