[Math] Solve equation with logarithm base 10

Question

I am going back to study log and unfortunately I don't know a lot.
I need to solve this:
$$
100= 10\log_{10} \left(50/x\right)
$$
I did the wrong calculation just moving stuff to the left, but I've been told is not right:
$$\begin{align}
100 \cdot x &= 10\log_{10} \cdot 50\\
x &= \left(10\log_{10} \cdot 50\right) / 100\\
x &= 0.016
\end{align}$$
I know that it is wrong, can someone explain me how to solve this?

Answer 1

$x\ne 0$ is part of the argument of $\log_{10},\,$ you cannot move it out the way you did. You can use $\log_{10}(50/x)= \log_{10}50 -\log_{10}x,\,$ or something like this: $$100 = 10\log_{10}\left(\frac{50}{x}\right)\quad (x\ne0)$$ $$\iff 10 = \log_{10}\left(\frac{50}{x}\right)$$ $$\iff 10^{10}=\frac{50}{x}$$ $$\iff x=\frac{50}{10^{10}}=\frac{1}{200000000}=5\times10^{-9}$$