Question

If a^2 + b^2 = 2ab, show that log a+b/2 = log a+ log b /2

Answer 1

If a²+b²=2ab therefore:

$\begin{gathered} a^2+b^2-2ab=0 \\ \Leftrightarrow(a-b)^2=0 \\ \Leftrightarrow a=b \end{gathered}$

Therefore:

$\log \frac{a+b}{2}=\log \frac{2a}{2}=\log a=\frac{2}{2}\log a=\frac{\log a+\log a}{2}=\frac{\log a+\log b}{2}$