Question

Confused over application of log power rule ?

The Question says

In x+In2x=12
I got the right answer by e^(2x^2)=e^12 which gives 218.39
But when I try it with log power rule,
Inx+In2x=12
In2x^2=12
2In2x=12
In2x=6
e^(In2x)=e^6,which gives me the wrong answer of 201.7. Why is this ?

Answer 1

Both your answers are wrong. See below.

Correct is `e^6 / sqrt 2 approx 285.27`

Explanation:

`ln x+ln2x=12`

I got the right answer by `color(red)(e)^(2x^2)=e^12` which gives 218.39

No, red bit is wrong

`ln x+ln2x=12`
`implies ln 2x^2=12`
`implies e^(ln 2x^2)=e^12`
`implies 2x^2=e^12` NOT `e^(2x^2)=e^12`
`implies x^2=e^12 /2`
`implies x=(e^12 /2)^(1/2) = e^6 / sqrt 2`

But when I try it with log power rule,
`lnx+ln2x=12`
`ln2x^2=12`
`2ln2x=12` Nah, sorry

Correct is
`ln2x^2=12`
`ln(sqrt 2 x)^2=12`
`2 ln(sqrt 2 x)=12`
`ln(sqrt 2 x)=6`
`e^(ln(sqrt 2 x))=e^6`
`sqrt 2 x=e^6`
`x=e^6 / sqrt 2`