Find $\log_{12}{60}$ if there is given $\log_{6}30$=a and $\log_{15}24$=b

Question

Can someone help me to find $\log_{12}{60}$ if there is given $\log_{6}30$=a and $\log_{15}24$=b?

Answer 1

Hints: Fill in details

$$\begin{align*} &\log_{12}60=\frac{\log_660}{\log_612}=\frac{\log_65}{1+\log_62}+1\\{}\\ &\log_{15}24=\frac{\log_624}{\log_615}=\frac{1+2\log_62}{1+\log_65-\log_62}\\{}\\ &\log_630=1+\log_65\end{align*}$$

And now just substitute (why and how?):

$$\log_65=a-1\;,\;\;b(a-\log_62)=1+2\log_62\implies (b+2)\log_62=ab-1\;\ldots$$