1.

If `log(a^3-b^3)-log3=3/2(log a+log b),` find the value of `(a/b)^3+(b/a)^3.`

Answer» `log(a^3 - b^3) - log3 = 3/2( log a + log b)`
`log((a^3-b^3)/3) = 3/2(log ab)`
`log ((a^3-b^3)/3) = log (ab)^(3/2)`
`(a^3- b^3)/3 = (ab)^(3/2)`
`a^3 - b^3 = 3(ab)^(3/2)` eqn1
`(a/b)^3- 1 = 3(ab)^(3/2)`
`(a/b)^3 = 1 + 3(a/b)^(3/2)`
`1 - (b/a)^3 = (3(ab^(3/2)))/a^3`
`1 - 3(b/a)^(3/2) = (b/a)^3`
`(a/b)^3 + (b/a)^3 = 1 + 3(a/b)^(3/2) + 1 - 3(a/b)^(3/2)`
`= 2 + 3 ((a^3 - b^3)/(ab)^(3/2))`
`= 2 + 3 [(3(ab)^(3/2))/(ab)^(3/2)]`
`= 2 + 3xx 3`
`= 11`
Answer


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