Is this a property of log :
$$\log_{1/x^3} x = -\frac{1}{3} \log_{x} x$$
I was looking through the solution of a sum I was solving the the only way the steps made sense was if this took place. I am confused because I have never come across properties of the base of a logarithm either online or in any of my reference books. Please help me out.
This is the part of the solution where I think the above property was applied.
$$2^{-\log_{1/8}124} =2^{\log_{2}5}\, 2^{-1}$$
First $2^{-1}$ was separated and then for the remaining it was written as a cube and then $3$ was taken out and cancelled. Does such a transformation exist?
Just use the base change for logarithm: you have in general that $ \log_a b=\frac{\log_c b}{\log_c a}$, so $$\log_{1/x^3} x =\frac{\log_x x}{\log_{x} \frac{1}{x^3}}=-\frac{1}{3} \log_x x$$ Having used the other properties: $\log u^v = v \log u$ and $\log \frac{1}{w}=-\log w$.