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If a ge b gt 1, then find the largest p...

If ` a ge b gt 1`, then find the largest possible value of the expression ` log_(a)(a//b)+log_(a)(b//a)`.

Text Solution

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Let ` x = log_(a) (a//b) + log_(b) (b//a)`
` = log_(a) a-log_(a)b+log_(b) b-log_(b) a`
` = 2- (log_(b) a+log_(a)b)`
` =- (sqrt(log_(b) a)-sqrt(log_(a)b))^(2) le 0`
Hence, the maximum value is 0.
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