# Help Evaluating Logarithm

1. Feb 22, 2012

### theintarnets

1. The problem statement, all variables and given/known data
Evaluate without a calculator:
(log34 + log29)2 - (log34 - log29)2

2. Relevant equations

3. The attempt at a solution
(log34 + log29)2 - (log34 - log29)2

(2log32 + 2log23)2 - (2log32 - 2log23)2

And now I'm stuck....

Last edited: Feb 22, 2012
2. Feb 22, 2012

### rock.freak667

You went from a minus to a divide, I think you confused it with log(a/b)= loga-logb

Try using this fact a2-b2 = (a+b)(a-b).

3. Feb 22, 2012

### Karamata

$$a^2-b^2=(a-b)(a+b)$$

EDIT: how to delete message

4. Feb 22, 2012

### eumyang

It also looks like you went from
(log34 + log29)2
to
(log342 + log292),
which is not true.

5. Feb 22, 2012

### theintarnets

Ohhhhh, I see. I shall re-attempt now. Thanks.

6. Feb 22, 2012

### Joffan

Hmmm. I think it's simpler than that, guys -
Setting A = log34 and B = log29

(A+B)2 - (A-B)2 = (A2 + 2AB + B2) - (A2 - 2AB + B2)

...

and later on using that lognxk = k.lognx
and that logab $\times$ logbc = logac

7. Feb 22, 2012

### theintarnets

Erm...Nevermind. I'm still stuck. Can someone walk me through it please?

Edit: I totally forgot about factoring. I'll try that, thanks!!

8. Feb 22, 2012

### Staff: Mentor

try this (A+B)^2 - (A-B)^2 = X^2 - Y^2 = ( X + Y ) ( X - Y )

where X=(A+B) and Y=(A-B)

and you get (A+B)^2 - (A-B)^2 =(2A) (2B) = 4AB

9. Feb 22, 2012

### theintarnets

Okay one of my friends just told me to try changing the base. I did that, and now I have a giant mess on my hands. I have

(ln2*ln4 + ln3*ln9 / ln3*ln2)^2 - (ln2*ln4 - ln3*ln9 / ln3*ln2)^2 and somehow I'm supposed to get 16 from all of that. I'm not really sure how...

10. Feb 22, 2012

### eumyang

No, no, NO! Don't change the base at the beginning. Do what Joffan suggested. You can change the base much later if you really want to.

11. Feb 22, 2012

### theintarnets

Okie. Trying again now.

Edit: I just can't seem to do it no matter what I try :(

Last edited: Feb 22, 2012
12. Feb 22, 2012

### eumyang

Show us what you have so far.

13. Feb 22, 2012

### theintarnets

Well I tried applying this:
Setting A = log34 and B = log29
(A+B)2 - (A-B)2 = (A2 + 2AB + B2) - (A2 - 2AB + B2)
The A2's and B2's will cancel out leaving me with 4AB which would be

4(log34 * log29)

The answer in the book says I'm supposed to get 16. I tried changing the base to get
4(ln4*ln9 / ln3*ln2)
I think I could maybe do 4(2ln2*2ln3 / ln3*ln2) but I'm not sure if that's correct. I think then maybe the ln2's and ln3's would cancel out leaving me with 4(2*2) which would be 16. But I'm not sure if that's correct.

14. Feb 22, 2012

### eumyang

That's correct.

15. Feb 22, 2012

### theintarnets

Yay!!! Thanks so much everyone!

16. Feb 22, 2012

### Joffan

Also, if you didn't want to change base,
\begin{align} log_34 \times log_29 & = log_32^2 \times log_23^2\\ &= 2log_32 \times 2log_23\\ &= 4(log_32.log_23)\\ &= 4(log_33) \\ &=4 \end{align}