Log_(3)(x) + log_(9)(x) = 12 solve for x​

Question

`log_(3)(x) + log_(9)(x) = 12`
solve for x​

Answer 1

x = 3⁸

Explanation:

Step(i):-

Given that

`log _(3) (x)+ log_(9) (x) =12`

`log _(3) (x)+ log_(3^(2) ) (x) =12`

we know that

`log^(a) _(b) = (loga)/(logb)`

`log _(3) (x)+ (logx)/(log3^(2) ) =12`

Step(ii):-

Apply log xⁿ = nlogx

`log _(3) (x)+(1)/(2) (logx)/(log3 ) =12`

`log _(3) (x)+ (1)/(2) log_(3 ) (x) =12`

`log _(3) (x)+ log_(3 ) (x)^{(1)/(2) } =12` ( ∵ log xⁿ = nlogx)

Apply log(ab) = loga+logb

`log _(3) (x (x^{(1)/(2) }) =12`

`log _(3) ( (x^{(3)/(2) }) =12`

`(3)/(2) log _(3) ( x) =12`

`(1)/(2) log _(3) ( x) = 4`

`log _(3) ( x) = 8`

we know that
`log _(b) ( x) = a` ⇒ x = bᵃ

∴ x = 3⁸