# Solving a simple logarithm

1. Jul 16, 2007

### obsolete

Hi everyone.

I know this question is quite simple but I can't wrap my head around it at the moment..

Solve for y: ln(x) + ln(y) = 0

I've tried differentiating both terms and then arranging for y but I get y = -x. The answer is meant to be y = 1/x.

Thanks all!

2. Jul 16, 2007

### ranger

ln(x) + ln(y) = ln(xy)

3. Jul 16, 2007

### Gib Z

Taking ln x to the other side and exponentiating gives:

$$y = e^{-\ln x}$$

I think the problem you are having is that you are thinking "$e^{\ln x}= x$, so this one should be -x". But remember the the negative sign is in the exponent.

If that is still not clear enough perhaps this will help : $a \log x = \log (x^a)$ for any base.

4. Jul 16, 2007

### Kummer

You cannot differentiate both sides, because those are numbers not functions. The derivatives will yield zero and nothing important.

5. Jul 16, 2007

### HallsofIvy

Staff Emeritus
Gosh, do I feel dumb! I did that by writing it as ln x= -ln y.

Yes, ln x+ ln y= ln xy= 0 is easier.

6. Jul 16, 2007

### obsolete

Thanks for the replies everyone.

I had already considered ln(x) + ln(y) = ln(xy) but I don't know how to approach the problem using this law. Any directions?

Thanks again.

7. Jul 16, 2007

### Dick

You've got ln(xy)=0. What then is the value of xy?

8. Jul 16, 2007

### K.J.Healey

Forget the ln(xy), and use your ln(x)+ln(y)=0
do what you did next, your gut instinct was right.
ln(x)=-ln(y)
now dont exponentiate yet, whats -ln(y)? Remeber a*log(x) = log(x^a) right? so whats (-1)*ln(y)?

After you understand that, exponentiate, and solve :) e^log(a)=a

9. Jul 16, 2007

### K.J.Healey

Just to add, the ln(xy) way works too. Its just that I don't want you to think that the way you started was wrong.

10. Jul 16, 2007

### obsolete

I finally understand =)

Thanks all!