I'm stuck on how to work with "e" and "ln" in certain ways. This is for a calculus project about track and field statistics and growth constants.

I'll post the longer stuff later in the post.

edit: you don't have to read the long part. offering help on the short part right below will be fine. thanks =)

So here's the quick end of things:

Known variables:

Vn1, Vo1, Vn2, Vo2, t1, t2

The unknown variable is A.

how can I simplify this. I'm basically trying to get rid of the "ln".

ln[(Vn1-A)/(Vo1-A)] = (t1/t2)*ln[(Vn2-A)/(Vo2-A)

I tried doing "e" raised to everything on both sides, so it looks like this:

e^{ln[(Vn1-A)/(Vo1-A)]} = e^{(t1/t2)*ln[(Vn2-A)/(Vo2-A)}

And that was when I got stuck. I know I can't split the right side into two parts. For example, I can't do:

e^{(t1/t2)} * e^{ln[(Vn2-A)/(Vo2-A)}

What do I do then?

OK, here's the longer part, just to check if I did anything wrong to begin with. The original problem looks like this. It's two equations to begin with. Variable "k" cancels out later, as I'll show.

Vn1 = A + (Vo1 - A)*e^(k*t1)

and

Vn2 = A + (Vo2 - A)*e^(k*t2)

Now I simplify each equation, and get this:

[(Vn1-A)/(Vo1-A)] = e^(k*t1)

and

[(Vn2-A)/(Vo2-A)] = e^(k*t2)

From here, I multiplied everything by ln, and got:

ln[(Vn1-A)/(Vo1-A)] = k*t1

and

ln[(Vn2-A)/(Vo2-A)] = k*t2

When I do systems of equations, I can cancel out the "k", so it becomes:

( ln[(Vn1-A)/(Vo1-A)] / ln[(Vn2-A)/(Vo2-A)] ) = (t1/t2)

After that, I can either rearrange the equation and get the equation I posted at the beginning of the post, or I can rearrange and get this:

ln[(Vn1-A)/(Vo1-A)] / t1 = ln[(Vn2-A)/(Vo2-A)] / t2

I'm not sure which way works better, but either way, i'm trying to still find "A".

Any way you could help would be appreciated. Thanks,

-Dan