How can I isolate a variable in a logarithmic equation?

Question

Sorry for such a simple question.

I have $y\ln(5) = 2\ln 3$

And I wanted to know if I can solve for $y$ by just dividing the logarithm on both sides?

So I would get $y = \frac {2\ln 3}{\ln(5)} = 2 \ln 3 - \ln 5$

Answer 1

To more directly answer your question, yes, you can divide both sides by $\ln(5)$ to isolate $y$.

However also as pointed out in other posts, you're mixing up properties. $\ln(a/b) = \ln(a)-\ln(b)$ - it's not $\ln(a)/\ln(b)=\ln(a)-\ln(b)$. For an easy way to see this, take $a=b=1$. Then

$$\ln\left( \frac 1 1 \right) = \ln(1) = 0 = 0 - 0 = \ln(1) - \ln(1)$$

but

$$\frac{\ln(1)}{\ln(1)} = \frac 0 0 \ne \ln(1) - \ln(1) = 0$$

Answer 2

You can only use that $$\ln\left(\frac{a}{b}\right)=\ln(a)-\ln(b)$$ for $$a,b>0$$

Answer 3

Note clearly the difference between $\frac {log (a)} {log(b)} $and $log (\frac a b )$: $\frac {log (a)} {log(b)} ≠log (\frac a b )= log (a)-log(b)$