How solve the given logarithmic problem

Question

Let

$$\log_{12}(18) = a$$

Then $$\log_{24}(16)$$ is equal to what in terms of a?

Answer 1

We have: $12^a = 18 \implies 2^{2a}\cdot 3^a = 2\cdot 3^2\implies 2a\ln 2+ a\ln 3 = \ln 2+2\ln 3$. Thus $b = \log_{24}16\implies 24^b = 16 = 2^4\implies 2^{3b}\cdot 3^b = 2^4\implies 3b\ln 2+b\ln 3=4\ln2\implies 3b+b\dfrac{\ln3}{\ln2}=4$. You can divide by $\ln2$ the first equation, and solve for $\dfrac{\ln3}{\ln2}$, then substitute it into the second equation to solve for $b$ in term of $a$.