Need to simplify a logarithmic expression

Question

Can someone simplify this ($\log$ here refers to the common logarithm)? $$\sqrt{4\log2+(\log5)^2} + \sqrt{4\log5+(\log2)^2}$$ I know this has a simple solution but I cannot find it.

Answer 1

It seems pretty sneaky. So I'm guessing those are common logs. Let's take a look at the first square root. We have

$$\sqrt{4\log2+(\log5)^2}=\sqrt{4(\log 10-\log5)+(\log5)^2}=\sqrt{4-4\log5+(\log5)^2}=2-\log 5$$

Similarly, the second square root yields $2-\log 2$. Sum them together to get $4-\log10=3$.