Question

Calculate ${\log }_{5}9.8$ in terms of $a$ and $b$, if $\log 2=a$ and $\log 7=b$.

Answer 1

${\log }_{5}9.8=\frac{\log 9.8}{\log 5}=\frac{\log \frac{98}{10}}{\log \frac{10}{2}}=\frac{\log 98-\log 10}{\log 10-\log 2}$

$=\frac{\log (49\times 2)-1}{1-\log 2}$

Given: $\log 2=a$ and $\log 7=b$

$\Rightarrow {\log }_{5}9.8=\frac{\left(\log 7^2\right)+\left(\log 2\right)-1}{1-\log 2}$

$=\frac{2\log 7+\log 2-1}{1-\log 2}$

$=\frac{2b+a-1}{1-a}$.