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Compute log(ab)(root(3)a//sqrtb)" if " ...

Compute ` log_(ab)(root(3)a//sqrtb)" if " log_(ab) a = 4`.

Text Solution

Verified by Experts

The correct Answer is:
`17/16`
`log_(ab) a = 4`
` or 1/(log_(a)ab) = 4`
` or 1/(log_(a) a+log_(a) b) = 4`
` or 1+log_(a) b = 1/4`
` or log_(a) b=- 3/4`
Now` log_(ab) (root(3)a//sqrtb)=(log(root(3)a//sqrtb))/(log ab) = (1/3loga - 1/2 log b)/(log a+ log b) `
` (1/3-1/2 (logb)/(log a))/(1+(log b)/(log a))`
` = (1/3-1/2 log_(a) b)/(1+ log_(a) b)`
`= (1/3+1/2*3/4)/(1-3/4)`
` =(17/24)/(1/4)=17/6`
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