# Simplifying an expression with e

• Mar 20th 2011, 06:46 PM
lfroehli
Simplifying an expression with e
Simplify the expression $\displaystyle e^{(\ln3) +2 \ln x}$.

My work (I feel like it's not legal!):

$\displaystyle \ \ln(e^{ln3+2 \ln x})$
$\displaystyle = \ln3+2 \ln x$
$\displaystyle = 2\ln (3x)$
$\displaystyle = \ln (3x^2)$
$\displaystyle = e^{ln3x^2}$
$\displaystyle = 3x^2$

Am I allowed to just take the natural log of an expression when simplifying since there's no other side to take the natural log of? Same question for bumping the expression up into the exponent of e?
• Mar 20th 2011, 06:49 PM
topsquark
Quote:

Originally Posted by lfroehli
Simplify the expression $\displaystyle e^{\ln3+2\lnx}$.

My work (I feel like it's not legal!):

$\displaystyle \ \ln(e^{ln3+2\lnx})$
$\displaystyle = \ln3+2\lnx$
$\displaystyle = 2\ln3x$
$\displaystyle = \ln3x^2$
$\displaystyle = e^{ln3x^2}$
$\displaystyle = 3x^2$

Am I allowed to just take the natural log of an expression when simplifying since there's no other side to take the natural log of? Same question for bumping the expression up into the exponent of e?

No, you can't.

$\displaystyle \displaystyle e^{ln(3) + 2} = e^{ln(3)} \cdot e^2$

Does this help?

-Dan

Wait. where did the x come from?
• Mar 20th 2011, 06:55 PM
lfroehli
ah, I messed up the question. It's simplify the expression $\displaystyle e^{\ln3 + 2 \ln{x}}$.
• Mar 20th 2011, 06:55 PM
mr fantastic
Quote:

Originally Posted by topsquark
No, you can't.

$\displaystyle \displaystyle e^{ln(3) + 2} = e^{ln(3)} \cdot e^2$

Does this help?

-Dan

Wait. where did the x come from?

The OP did not use the latex commands correctly and so the x did not show up. I have fixed this.
• Mar 20th 2011, 06:58 PM
Prove It
First simplify the exponent into a single logarithm...

$\displaystyle \displaystyle \ln{(3)} + 2\ln{(x)} = \ln{(3)} + \ln{\left(x^2\right)} = \ln{(3x^2)}$.

Then $\displaystyle \displaystyle e^{\ln{(3)} + 2\ln{(x)}} = e^{\ln{\left(3x^2\right)}}$. The rest of the solution should be obvious...
• Mar 21st 2011, 05:32 AM
HallsofIvy
Quote:

Originally Posted by lfroehli
Simplify the expression $\displaystyle e^{(\ln3) +2 \ln x}$.

My work (I feel like it's not legal!):

$\displaystyle \ \ln(e^{ln3+2 \ln x})$
$\displaystyle = \ln3+2 \ln x$

This is alright if you remember that you took the log and reverse it at the end.

Quote:

$\displaystyle = 2\ln (3x)$
But this is NOT right. The 2 is multiplying only ln(x), not log 3. What you need to do is take that 2 inside the ln:
$\displaystyle ln 3+ 2ln x= ln 3+ ln x^2= ln (3x^2)$

Quote:

$\displaystyle = \ln (3x^2)$
Okay, now you have fixed the "2" problem!

Quote:

$\displaystyle = e^{ln3x^2}$
$\displaystyle = 3x^2$
Good! Now you have reversed the ln you did at the beginning and have the correct answer.

Quote:

Am I allowed to just take the natural log of an expression when simplifying since there's no other side to take the natural log of? Same question for bumping the expression up into the exponent of e?
What you did is correct but it would have been simpler to use the logarithm properties in the exponent.
$\displaystyle e^{ln(3)+ 2ln(x)}= e^{ln(3)+ ln(x^2)}= e^{ln(3x^2)}= 3x^2$.