If $2^a=3^b$ find $\frac{a}{b}$

Question

I tried many different things but still couldn't solve it. Could you please give me a clue?

Answer 1

Try taking the natural log of both sides. Remember that $\ln 2^a=a\ln 2$ and $\ln 3^b=b\ln 3$.

$$a\ln 2=b\ln 3$$

Now, you can use division to solve for $\frac a b$.

Answer 2

$2^a = 3^b; \tag 1$

$\ln (2^a) = \ln (3^b); \tag 2$

$a \ln 2 = b \ln 3; \tag 3$

$\dfrac{a}{b} = \dfrac{\ln 3}{\ln 2}. \tag 4$