CAT 2024 Slot 2 QA Question 1: If $a$, $b$ and $c$ are positive real numbers such that $a 10 \geq b \geq c$ and $\dfrac{\log_8(a+b)}{\log_2 c} + \dfrac{\log_{27}(a-b)}{\log_3 c} = \dfrac{2}{3}$, then the greatest possible integer

Question

If $a$, $b$ and $c$ are positive real numbers such that $a > 10 \geq b \geq c$ and $\dfrac{\log_8(a+b)}{\log_2 c} + \dfrac{\log_{27}(a-b)}{\log_3 c} = \dfrac{2}{3}$, then the greatest possible integer

Accepted Answer

Convert both logarithmic fractions to logarithms with base $c$.
$\dfrac{\log_8(a+b)}{\log_2 c} = \dfrac{1}{3}\log_c(a+b)$
$\dfrac{\log_{27}(a-b)}{\log_3 c} = \dfrac{1}{3}\log_c(a-b)$
$\dfrac{1}{3}[\log_c(a+b)+\log_c(a-b)] = \dfrac{2}{3}$
$\Rightarrow \log_c((a+b)(a-b)) = 2$
$\Rightarrow a^2 - b^2 =