Logarithm-question: Using $\log_{24}(12)$ , find $\log_{24}(6)$

Question

Logarithm-question:

Using $\log_{24}(12)$, find $\log_{24}(6)$

Thanks.

Answer 1

Let $\log_{24}12=x.$

Then $\log_{24}2=\log_{24}24-\log_{24}12=1-x.$

So $\log_{24}6=\log_{24}12-\log_{24}2=2x-1.$

Here's verification:

$(\log_{24}12)\cdot 2-1=\log_{24}12+\log_{24}12-\log_{24}24=\log_{24}(\frac{12\cdot 12}{24})=\log_{24}6.$