stylaria3y

2022-07-21

What log rule was used to simply this expression?
I'm unclear how the left side is equal to the right side.
$365\mathrm{log}\left(365\right)-365-305\mathrm{log}\left(305\right)+305-60\mathrm{log}\left(365\right)=305\mathrm{log}\left(\frac{365}{305}\right)-60$
I know $\mathrm{log}\left(a\right)-\mathrm{log}\left(b\right)=\mathrm{log}\left(a/b\right)$ 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 & Explanation

There are a couple of steps missing.

Makenna Booker

Collect the constants (-365 + 305 = -60), and the terms with $log\left(365\right)$
$365\mathrm{log}\left(365\right)-365-305\mathrm{log}\left(305\right)+305-60\mathrm{log}\left(365\right)=305\mathrm{log}\left(365\right)-305\mathrm{log}\left(305\right)-60$
Now factor out 305, and use the identity you mentioned:
$305\left(\mathrm{log}\left(365\right)-\mathrm{log}\left(305\right)\right)-60=305\mathrm{log}\left(365/305\right)-60$

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