How do I simplify this Log with a Fraction in it?

Question

So I have:

$$ \log_2(5x) + \log_2 3 + \frac{\log_2 10}{2} $$

I understand that when there is addition, and the bases are the same, I can simply multiply what is in the parenthesis. So for the first part, I'd get $\log_2(15x)$. I'm stuck now, because I'm not sure what to do with the third log term, since the entire thing is being divided by two.

Answer 1

You may write $$ \log_2(15x)+\frac12 \log_2(10)=\log_2(15x)+\log_2(10^{1/2})=\log_2(15x \cdot10^{1/2}). $$

Answer 2

$\frac{\log_2 10}{2}=\frac{1}{2} \log_2 10 = \log_2 {10^{1/2} = \log_2 \sqrt{10}}$. Combining this term with the previous terms gives the answer $\log_2(15x \cdot \sqrt{10})$.