Confused over application of log power rule ?

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 ?

1 Answer

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Answer 1 · 2016-08-14T22:05:11

Both your answers are wrong. See below.

Correct is $\frac{e^{6}}{\sqrt{2}} \approx 285.27$ $e^6 / sqrt 2 approx 285.27$

Explanation:

$ln x + ln 2 x = 12$ $ln x+ln2x=12$

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

No, red bit is wrong

$ln x + ln 2 x = 12$ $ln x+ln2x=12$
$\Rightarrow ln 2 x^{2} = 12$ $implies ln 2x^2=12$
$\Rightarrow e^{ln 2 x^{2}} = e^{12}$ $implies e^(ln 2x^2)=e^12$
$\Rightarrow 2 x^{2} = e^{12}$ $implies 2x^2=e^12$ NOT $e^{2 x^{2}} = e^{12}$ $e^(2x^2)=e^12$
$\Rightarrow x^{2} = \frac{e^{12}}{2}$ $implies x^2=e^12 /2$
$\Rightarrow x = {(\frac{e^{12}}{2})}^{\frac{1}{2}} = \frac{e^{6}}{\sqrt{2}}$ $implies x=(e^12 /2)^(1/2) = e^6 / sqrt 2$

But when I try it with log power rule,
$ln x + ln 2 x = 12$ $lnx+ln2x=12$
$ln 2 x^{2} = 12$ $ln2x^2=12$
$2 ln 2 x = 12$ $2ln2x=12$ Nah, sorry

Correct is
$ln 2 x^{2} = 12$ $ln2x^2=12$
${ln (\sqrt{2} x)}^{2} = 12$ $ln(sqrt 2 x)^2=12$
$2 ln (\sqrt{2} x) = 12$ $2 ln(sqrt 2 x)=12$
$ln (\sqrt{2} x) = 6$ $ln(sqrt 2 x)=6$
$e^{ln (\sqrt{2} x)} = e^{6}$ $e^(ln(sqrt 2 x))=e^6$
$\sqrt{2} x = e^{6}$ $sqrt 2 x=e^6$
$x = \frac{e^{6}}{\sqrt{2}}$ $x=e^6 / sqrt 2$

Science

Math

Humanities

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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

Explanation:

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Impact of this question

Science

Math

Humanities

... and beyond

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

Explanation:

Related questions

Impact of this question

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 ?