# Thread: Giving an Exact Solution to a Logarithm

1. ## Giving an Exact Solution to a Logarithm

Subject are exactly what the instructions are and I have no problem doing logarithms until I came upon this problem. Teacher didn't really give any examples like this one and I am totally lost.

I have no idea how to write that exactly as I see it
Any help is GREATLY appreciated thank you!

2. ## Re: Giving an Exact Solution to a Logarithm

\displaystyle \displaystyle \begin{align*} \log_2{\sqrt{3x^2}} &= \frac{3}{2} \\ \sqrt{3x^2} &= 2^{\frac{3}{2}} \end{align*}

Go from here.

3. ## Re: Giving an Exact Solution to a Logarithm

I tried that earlier and just got stuck again as I've never done a logarithm like this.

Am I on the right track? If so I have no idea what to do next =(

4. ## Re: Giving an Exact Solution to a Logarithm

The answer I got is X is approximately .9428

5. ## Re: Giving an Exact Solution to a Logarithm

Originally Posted by Mollers
The answer I got is X is approximately .9428
"approximately" isn't exact though is it
This problem is completely solvable without a calculator. No decimals!

6. ## Re: Giving an Exact Solution to a Logarithm

$\displaystyle \log_2(\sqrt{3x^2})=\frac{3}{2}$

$\displaystyle \frac{1}{2} \log_2(3x^2)=\frac{3}{2}$

$\displaystyle \log_2(3x^2)=3$

$\displaystyle 3x^2=2^3$

$\displaystyle \vdots$

7. ## Re: Giving an Exact Solution to a Logarithm

I'm using the fact that log(sqrty)=1/2logy
So from your question we get 1/2log3x^2=3/2 log3x^2=3 So 3x^2=2^3 (since log base is 2)
So x^2=8/3 x=sqrt(8/3) =root8/root3 =2root2/root3
Multiply top and bottom by root3 Get x=2root6/3

8. ## Re: Giving an Exact Solution to a Logarithm

First off thank you all for the help I couldn't get no where near where I am in this problem without you. I am stuck again as I cannot figure out how to write the last part of what Biffboy said..."root8/root3 =2root2/root3 --------> Multiply top and bottom by root3 Get x=2root6/3"

This is where I get stuck.

9. ## Re: Giving an Exact Solution to a Logarithm

ignore the bottom thumbnail

10. ## Re: Giving an Exact Solution to a Logarithm

Originally Posted by Mollers
First off thank you all for the help I couldn't get no where near where I am in this problem without you. I am stuck again as I cannot figure out how to write the last part of what Biffboy said..."root8/root3 =2root2/root3 --------> Multiply top and bottom by root3 Get x=2root6/3"

This is where I get stuck.
Here we go:

$\displaystyle 3x^2=8$

$\displaystyle x^2=\frac{8}{3}$

$\displaystyle x=\sqrt{\frac{8}{3}}$

$\displaystyle x=\frac{\sqrt{8}}{\sqrt{3}}$

$\displaystyle x=\frac{2\sqrt{2}}{\sqrt{3}}$

"Rationalize" the denominator by multipling numerator and denominator by $\displaystyle \sqrt{3}$

$\displaystyle x=\frac{2\sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}=\frac{2\sqrt{6}}{3}$

11. ## Re: Giving an Exact Solution to a Logarithm

thank you so much! I dont understand why I was given this problem when I was never given an example that was anywhere near similar.

12. ## Re: Giving an Exact Solution to a Logarithm

I have been looking at this problem more and thinking about it and wanted to make sure about something.

Since the directions are to give an exact solution would I leave the answer as 2 sqaure root 6/3 or should I compute this and get a decimal answer?

13. ## Re: Giving an Exact Solution to a Logarithm

Leave it as it is. Any decimal answer you got would be an approximation.