TITLE: If $\log_{12}54=a$ then $ \log_{6}12=?$
QUESTION [5 upvotes]: I am given $$\log_{12}54=a$$
So what will be value of $ \log_{6}12?$
I used base changing theorem and wrote expression as 
$$\frac{\log_{6}54}{ \log_{6}12} =a$$
And then $$ \frac{1+\log_{6}9}{ a} = \log_{6}12$$ now what to do?

REPLY [2 votes]: $\bf hint:$
decompose the numbers $54 = 2*3^3, 12 = 2^2*3, 6 =2*3$ as use the change of basis formula $\log_a(b) = \dfrac{\ln b}{\ln a}.$ so we get 
$$a = \dfrac{\ln 2 + 3 \ln 3}{2\ln 2 + \ln 3}, b = \dfrac{2\ln 2 + \ln 3}{\ln 2 + \ln 3} $$
observe that both $a$ and $b$ depend only on the ratio $\dfrac{\ln 2}{\ln 3}.$ eliminating the ratio should give you a relation between $a$ and $b.$