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If a,b, and c are positive and 9a+3b+c=9...

If a,b, and c are positive and `9a+3b+c=90`, then the maximum value of `(log a+log b+log c)` is (base of the logarithm is 10)________.

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The correct Answer is:
3
`9a + 3b + c = 90`
`implies 3a + b + (c )/(3) = 30`
Now consider number 3a, b and `(C )/(3)`
`(3a + b xx (c )/(3))^((1)/(3)) le (3a + b + (c )/(3))/(3)` (as G.M `ge` AM)
or `(abc)^(1//3) le (30)/(3) = 10`
or `abc le 1000`
`implies log a + log b + log c le 3`
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