Question

Let $L$ denotes the value of a satisfying the equation ${\log }_{\sqrt{8}}(a)=\frac{10}{3}$ and $M$ denotes the value of $b$ satisfying the equation $4^{{\log }_{9}3}+9^{{\log }_{2}4}=10^{{\log }_{b}83}$. Find the sum of the digits in $(L+M)$

Answer 1

Answer: 6

$L:{\log }_{\sqrt{8}}(a)=\frac{10}{3}\Rightarrow a=(\sqrt{8}{)}^{\frac{10}{3}}\Rightarrow a={\left(2^3\right)}^{\frac{10}{2\times 3}}=2^5=32$
$M:4^{{\log }_{9}3}+9^{{\log }_{2}4}=10^{{\log }_{b}83}$
$\Rightarrow 2+81=10^{{\log }_{b}83}\Rightarrow 83=10^{{\log }_{b}83}$
$\therefore 83=(83{)}^{{\log }_{b}10}\Rightarrow {\log }_{b}10=1\Rightarrow b=10=M$
$\text{ Hence, }(L+M)=(a+b)=32+10=42$