# How do you solve log516 - log52t = log52?

Nov 15, 2015

Remember the logarithm rule: ${\log}_{a} \left(\frac{x}{y}\right) = {\log}_{a} x - {\log}_{a} y$
You can work in reverse:
$\log 516 - \log 52 t = \log 52$
$\log \left(\frac{516}{52 t}\right) = \log 52$

Now, there are multiple things you could do from here, but the easiest would be to recognize that if $\log a = \log b , a = b$.
Therefore, $\frac{516}{52 t} = 52$.
We can solve for $t$.
$516 = 2704 {t}^{2}$
$\frac{516}{2704} = {t}^{2}$
$\sqrt{\frac{516}{2704}} = t$
$\sqrt{\frac{129}{676}} = t$

Notice that we only use the positive square root, excluding $- \sqrt{\frac{129}{676}}$. We must do this since if we plugged$- \sqrt{\frac{129}{676}}$ into $\log 52 t$, we would get a negative number, which is impossible.