[Math] Simplify $\frac{5}{6}\log\left(\frac{5}{4}\right) – \frac{1}{6}\log(2)$ to $\log\left(\frac{5}{4}\right) – \frac{1}{6}\log\left(\frac{5}{2}\right)$

Question

I'm trying to bring this expression:

$$\frac{5}{6}\log\left(\frac{5}{4}\right) – \frac{1}{6}\log(2)$$

To this one:

$$\log\left(\frac{5}{4}\right) – \frac{1}{6}\log\left(\frac{5}{2}\right)$$

Where $\log$ is the natural algorithm.

I know the two expressions are equal (checked with wolfram) but I really can't find the correct passages… Could you please help me?

Answer 1

$\frac56\log\left(\frac54\right)-\frac16\log(2)=\log\left(\frac54\right)-\frac16\log\left(\frac54\right)-\frac16\log(2)$

$= \log\left(\frac54\right)-(\frac16\log\left(\frac54\right)+\frac16\log(2))$

$=\log\left(\frac54\right)-\frac16\log\left(\frac{5\cdot2}{4}\right)$

$=\log\left(\frac54\right)-\frac16\log\left(\frac52\right)$

I should add: Subtracting logs is equivalent to division. The reason for potential problems here is $\frac{\frac{a}{b}}c = \frac{a}{b\cdot c}$ and not $\frac{a}{(\frac{b}{c})}$