Question

If ${\log }_{3}5=x$ and ${\log }_{25}11=y$ then the value of ${\log }_3\left(\frac{11}{3}\right)$ in terms of x and y is

Answer 1

Given, ${\log }_{3}5=x$

$\Rightarrow \frac{\log 5}{\log 3}=x$ (equation $1$)

And, ${\log }_{25}11=y$

$\Rightarrow \frac{\log 11}{2\log 5}=y$ (equation $2$)

From $1$ and $2$

$\Rightarrow \frac{\log \left(11\right)}{2x\log 3}=y$

$\Rightarrow {\log }_3\left(11\right)=\left(2xy\right)$

$\Rightarrow {\log }_3\left(11/3\right)={\log }_3\left(11\right)-1=2xy-1$