<s>[INST] Solve x in logarithm equation I am trying to solve $x$ for $2log_{10} (x-4) - log_{10}4(x-1) = 0$
I have the key with the answer 10 and have confirmed this is correct using Wolfram Alpha but which steps should I take to reach that answer?
 [/INST] Start with original problem:
$$2 \log_{10} (x-4) - \log_{10}4(x-1) = 0$$
A rule of logarithm: $2 \log_{10} (x-4)=\log_{10}(x-4)^2$:
$$ \log_{10} (x-4)^2 - \log_{10} (4x-4)= 0$$
Another rule of logarithm: Express as the left side of the equation as a single product: 
$$ \log_{10} \frac{(x-4)^2}{4x-4}= 0$$
Convert to exponential form:
$$ 10^{\log_{10} \frac{(x-4)^2}{4x-4}}= 10^0$$
or 
$$\frac{(x-4)^2}{4x-4}=1$$
and when algebraically rearranging
$$x^2-8x+16=4x-4$$
or
$$x^2-12x+20=0$$
which is now a quadratic equation that you can solve.
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