Mastering Logarithms: Simplifying a Tricky Problem in Just a Few Steps

[Lancelot59]

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

Homework Statement

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

Homework Equations

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.

 

[Dr. Lady]

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

 

[Gauged]

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

 

[Lancelot59]

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

 

[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].[/itex]

 

[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;

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

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

$$9-x^2=5$$

 

[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?

 

[HallsofIvy]

What is the definition of "logarithm"?

 

[Lancelot59]

The inverse of an exponential function.

 

[tiny-tim]

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?

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.

log(3-x) + log(3+x) = log(5)
…
log(9-x^2) = log(5)

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

An equation like this, adding nothing but logs, will have the same result in any base of logs.

That's because log_ax = logx/loga …

so just divide that equation by log a and you get

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

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

 

[Lancelot59]

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

Thanks a lot! That really helped.

 

[Дьявол]

You need to use the properties of logarithms.

You can find them http://people.hofstra.edu/Stefan_Waner/Realworld/calctopic1/logs.html , and also some good examples.

In this particular case the property log_a(xy)= log_ax + log_ay should be used.

Regards.

 

[Lancelot59]

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

 

[tiny-tim]

… Although I'm still stuck on all the other questions.

Get some sleep :zzz: then start another thread