Let $a$ and $b$ be real numbers greater than $1$ for which there exists a positive real number $c$ , different from $1$ , such that $2 ({log}_{a} c + {log}_{b} c) = 9 {log}_{a} c$ . Find the largest possible value of ${log}_{a} b$ .

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Solution

Given that,
$2 ({log}_{a} c + {log}_{b} c) = 9 {log}_{a} c$
$∴ {log}_{a} b = 2 \times 9 {log}_{a} c$
$= 18 {log}_{a} c$

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