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

$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} $
$\Rightarrow 2+81=10^{\log _b 83} \Rightarrow 83=10^{\log _b 83}$
$\therefore 83=(83)^{\log _{ b } 10} \Rightarrow \log _{ b } 10=1 \Rightarrow b =10= M$
$\text { Hence, }( L + M )=( a + b )=32+10=42$