If it is given that $\frac{\log x}{\log y }=3/4$, can I write $\log_y x=3/4$?

Question

Change of base formula:

$$ \log_b x = \frac{\log_a x}{\log_a b}$$

So if it is given that $\frac{\log x}{\log y }=3/4$

Can I write $\log_y x=3/4$?

Answer 1

Yes, you have $$\frac {\log x}{\log y} = \log _{y}x = \frac {3}{4}$$

The base of logarithm in this case is $10$ but you can have it in any other base as well.

For example $$\frac {\ln x}{\ln y} = \log _{y}x = \frac {3}{4}$$ is also valid.

Answer 2

Yes. The way that "change of base" formula is derived is to use the fact that "$z= log_y(x)$" is equivalent to $x= y^z$. So saying $log_y(x)= \frac{3}{4}$ is the same as saying $x= y^{3/4}$. That, of course, is the same as $x^4= y^3$. Now, taking the logarithm, to some base b, of both sides $log_b(x^4)= 4log_b(x)= log_b(y^3)= 3log_b(y)$. From that, $\frac{log_b(x)}{log_b{y}}= \frac{3}{4}$.

To prove the other way, just reverse the above: From $\frac{log_b(x)}{log_b{y}}= \frac{3}{4}$ we have $log_b(x^4)= 4log_b(x)= log_b(y^3)= 3log_b(y)$ so that $x^4= y^3$, etc.