Question

The value of log3 5 x log25 9 is

Answer 1

$\log_{3}5\cdot\log_{25}9=\dfrac{1}{\log_{5}3}\cdot\log_{25}9=\dfrac{1}{\log_{5}3}\cdot\log_{5^2}9=\dfrac{1}{\log_{5}3}\cdot\dfrac{1}{2}\log_{5}9=\\\\\\= \dfrac{1}{\log_{5}3}\cdot\dfrac{1}{2}\log_{5}3^2=\dfrac{1}{\log_{5}3}\cdot\dfrac{2}{2}\log_{5}3=\dfrac{1}{\log_{5}3}\cdot\log_{5}3=\dfrac{\log_{5}3}{\log_{5}3}=\boxed{1}$

Answer 2

$\log_35\cdot\log_{25}9=\log_35\cdot\log_{25}3^2\\\\/use:\ \log x^n=n\log x/\\\\=\log_35\cdot2\log_{25}3\\\\/use:\log_ab=\dfrac{1}{\log_ba}/\\\\=\log_35\cdot2\cdot\dfrac{1}{\log_325}=\log_35\cdot\dfrac{2}{\log_35^2}=\dfrac{2\log_35}{2\log_35}=1$