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 ?

1 Answer
Eddie
Aug 14, 2016

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

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