Question

if a^2+b^2=7ab . Prove that : 2ln((a+b)/3)=lna+lnb.

Answer 1

\[2\ln\left(\frac{a+b}{3}\right)\]

\[=\ln\left(\left[\frac{a+b}{3}\right]^2\right)\]

\[=\ln\left(\frac{a^2+b^2+2ab}{3^2}\right)\]

\[=\ln\left(\frac{7ab+2ab}{9}\right)\]

\[=\ln\left(\frac{9ab}{9}\right)\]

\[=\ln\left(ab\right)\]

\[=\ln(a)+\ln(b)\]