Question

Calculate $lo{g}_9$ $40$ if $log15$ = a and $lo{g}_{20}$ 50 = b

Answer 1

$b={\log }_{20}50=\frac{\log 50}{\log 20}=\frac{\log \frac{100}{2}}{\log (2\times 10)}$

$⟹b=\frac{2-\log 2}{1+\log 2}$

$⟹b+b\log 2=2-\log 2$

$⟹\log 2=\frac{2-b}{b+1}$

$a=\log 15=\log 3+\log 5=\log 3+\log 10-\log 2=\log 3+1-\log 2$

$⟹log3=a+\log 2-1$

Now-

${\log }_{9}40=\frac{\log 40}{\log 9}=\frac{1+2\log 2}{2\log 3}$

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

$=\frac{5-b}{ab+a+1-2b}$