# Exponential Variable and Logarithms

1. Mar 5, 2013

### jsully

1. The problem statement, all variables and given/known data

I'm trying to find the value of a variable which happens to be an exponent.

2. Relevant equations

$230,000=1500\frac{(1+.00077)^n-1}{.00077}$

I believe I need to use logarithms to get to my answer, but I've reviewed logarithm rules and I'm stuck.

3. The attempt at a solution

I've divided 230k by 1500, but am stuck at that point.

I'm now at

$153.33=\frac{(1+.00077)^n-1}{.00077}$

...and stuck :( I think I need to log both sides, something like $log 153.33=(n)log \frac{(1+.00077)-1}{.00077}$
I've tried simplifying the right side, but I end up at 1 which means the right side ends up being zero, which doesn't make any sense.

Any assistance would be greatly appreciated.

2. Mar 5, 2013

### Fredrik

Staff Emeritus
Do you know how to solve $x^n=y$ for n? Can you rewrite the equation in that form?

3. Mar 5, 2013

### eumyang

No... you can't do that! Before taking the logarithm of both sides, I would isolate the (1+.00077)^n portion. Can you do that?

4. Mar 5, 2013

### jsully

Would it be $n=\frac{log153.33}{log(1+.00077)-1}{.00077}$

Last edited: Mar 5, 2013
5. Mar 5, 2013

### jsully

Yeah, nevermind that can't be right. Very frustrating..

6. Mar 5, 2013

### jsully

I mean, I know that if x^n=y then ln(y)/ln(x)=n. I can't figure out how to write using my values though.

7. Mar 5, 2013

### SteamKing

Staff Emeritus
No. In addition to reviewing the rules about logarithms, you need to brush up on your algebra as well.

You had 153.33 = ((1+0.00077)^n - 1) / 0.00077

You can further simplify:

153.33(0.00077) = (1+0.00077)^n - 1
153.33(0.00077)+1 = (1+0.00077)^n

Now use logarithms:

log (153.33(0.00077) + 1) = n log (1+0.00077)

Therefore:

n = log(153.33(0.00077)+1) / log (1+0.00077)

or

n = 144.99

8. Mar 5, 2013

### Fredrik

Staff Emeritus
You probably shouldn't give away the complete solution like that. A better hint would be to ask if jsully knows how to solve
$$a=b\frac{x-1}{c}$$ for x when a,b,c are non-zero real numbers.