# Thread: Exponent rules with logarithms

1. ## Exponent rules with logarithms

Hello,

I'm having trouble remembering a basic rule.

The question is:
(e^x)^2 = 3

My next step is:

ln(e^x)^2 = ln(3)

To my understanding, the ln and e knock each other out bringing down the x and 2. My question is, do I add them to make:

x+2 = ln(3) ? I'm not sure about the left side of that equation.

Thanks for any help!

2. Here's what I got...so unsure about it...

2x = ln(3)
x=1/2 ln (3)
x= 0.549

3. Originally Posted by shade
Here's what I got...so unsure about it...

2x = ln(3)
x=1/2 ln (3)
x= 0.549
Your calculations are OK. I personally prefer:

$\displaystyle x=\frac12 \cdot \ln(3) = \ln(\sqrt{3})$

4. Originally Posted by shade
Hello,

I'm having trouble remembering a basic rule.

The question is:
(e^x)^2 = 3

My next step is:

ln((e^x)^2) = ln(3)

To my understanding, the ln and e knock each other out bringing down the x and 2. My question is, do I add them? No
$\displaystyle (e^x)^2 = 3~\iff~ e^{2x} = 3~\implies~2x = \ln(3)$

Solve for $\displaystyle x\!:\;\;(e^x)^2 \:= \:3$
Take the square root of both sides: .$\displaystyle e^x \:=\:\sqrt{3}$

Take logs: .$\displaystyle \ln(e^x) \:=\:\ln(\sqrt{3}) \quad\Rightarrow\quad x\cdot\ln(e) \:=\:\ln(\sqrt{3})$

Since $\displaystyle \ln(e) = 1$, we have: .$\displaystyle x \:=\:\ln(\sqrt{3})$

6. Originally Posted by shade
Hello,

I'm having trouble remembering a basic rule.

The question is:
(e^x)^2 = 3

My next step is:

ln(e^x)^2 = ln(3)

To my understanding, the ln and e knock each other out bringing down the x and 2. My question is, do I add them to make:

x+2 = ln(3) ? I'm not sure about the left side of that equation.

Thanks for any help!
You are fine until this...

$\displaystyle ln(e^x)^2 = ln(3)$

Rules of logs say that the power ("2" in this case) gets brought down as a coefficient...

$\displaystyle 2ln(e^x) = ln(3)$

Now, cancel out the natural log with base e

$\displaystyle 2x = ln(3)$

Solve for x...

$\displaystyle x = \frac{ln(3)}{2} \Rightarrow ln(3^{\frac{1}{2}}) = ln(\sqrt{3})$

7. Thanks so much for everyone's help!