# Logarithims Question

1. Dec 13, 2008

### Lancelot59

Well this is my worst area. Not sure why though. Here's the latest problem I'm stuck with.

1. The problem statement, all variables and given/known data
log(3-x) + log(3+x) = log(5)

2. Relevant equations

3. The attempt at a solution
log(9-x^2) = log(5)

The answer is plus/minus 2, but I'm really not sure on where to go from here.

This is the stuck point.

2. Dec 13, 2008

Well, it looks like you can take 10^ of each side... see where that gets you.

Last edited: Dec 13, 2008
3. Dec 13, 2008

### Gauged

There is no need to change the base of the equation. Think of it as a basic polynomial when equal to zero.

Last edited: Dec 13, 2008
4. Dec 13, 2008

### Lancelot59

Could you give me an example? I'm not really sure what you mean by that.

5. Dec 13, 2008

### gabbagabbahey

Are you using Log to represent log base 10, or the natural log?

Assuming you are using the former, you are essentially given an equation $f(x)=g(x)$....If this equation holds, then so must the equation [itex]10^{f(x)}=10^{g(x)}[/tex]...Use that with f(x)=Log[9-x^2] and g(x)=Log[5].

6. Dec 13, 2008

### Gauged

Since you're raising the same base of 10 to some number x, you can treat the equation as if there were no logarithmic notation. Using the addition property of logarithms, and the fact that no change of base is needed;

$$\log(x-3)+\log(x+3)=\log(5)$$

$$\log(9-x^2)=\log(5)$$

$$9-x^2=5$$

Last edited: Dec 13, 2008
7. Dec 13, 2008

### Lancelot59

Sorry guys, I forgot to put in the "solve for x" part of the question. But this did enlighten me. For the longest time I thought that the only way to get a log was by using that method where there is a base and an exponent equalling another number.

My teacher didn't explain this very well.

From what I understand so far to use addition, and subtraction rules the logs need to be of the same base.

And to remove logs from certain numbers all logs in the equation need to have the same base.

Is this correct?

8. Dec 14, 2008

### HallsofIvy

Staff Emeritus
What is the definition of "logarithm"?

9. Dec 14, 2008

### Lancelot59

The inverse of an exponential function.

10. Dec 14, 2008

### tiny-tim

Hi Lancelot59!

Yes, logs in the same equation need to be of the same base (if they're not, you'll have to convert some of them).

If you're given an equation with several logs in, they will be of the same base.
An equation like this, adding nothing but logs, will have the same result in any base of logs.

That's because logax = logx/loga …

so just divide that equation by log a and you get

loga(3-x) + loga(3+x) = loga(5).

In other words, your solution log(9-x^2) = log(5) works in any base, and proves that 9-x^2 = 5.

11. Dec 14, 2008

### Lancelot59

So I can eliminate the logs from the equation and solve for x then. Awesome!

Thanks a lot! That really helped.

12. Dec 14, 2008

### Дьявол

You need to use the properties of logarithms.

You can find them here, and also some good examples.

In this particular case the property loga(xy)= logax + logay should be used.

Regards.

13. Dec 14, 2008

### Lancelot59

I put it together and managed to solve it. Although I'm still stuck on all the other questions.

14. Dec 14, 2008

### tiny-tim

Get some sleep :zzz: then start another thread