Compound Interest Formula and Natural Logarithms

[Gothika]

Homework Statement

Solve the compound interest formula for r by using natural logarithms.

Homework Equations

A=P(1+r/n)^nt

The Attempt at a Solution

1400 = 1000(1+r/360)^(360*2)

1.4 = (1+r/360)⁷²⁰

ln(1.4) = 720ln((360+r)/360)

I'm not sure where to go after this. Did I make a mistake?

 

[cepheid]

Welcome to PF Gothika!

You haven't done anything wrong so far. I would recommend continuing with steps to isolate r on one side of the equation. For instance, you'd divide both sides by 720, and then you'd be left with r in an expression inside a natural logarithm. So the only way to get at r would be to get rid of that natural logarithm by exponentiating both sides.

Can I also make a suggestion? Don't plug in numbers until the very end. Keep things in terms of A, P, r, n, and t, and just work it out algebraically. That way you've got an expression for 'r' that is true regardless of the specific value of the principal, interest rate, number of compounding periods etc. Furthermore, this dramatically reduces clutter and just makes things much cleaner.

 

[Gothika]

ln(1.4) = 720ln((360+r)/360)

I divided both sides by 720 and got:

ln(1.4)/720 = ln((360+r)/360)

But I'm not entirely sure what you mean by exponentiating both sides

 

[cepheid]

ln(1.4) = 720ln((360+r)/360)

I divided both sides by 720 and got:

ln(1.4)/720 = ln((360+r)/360)

But I'm not entirely sure what you mean by exponentiating both sides

I mean carry out the operation that is the direct inverse of taking a natural logarithm.

If I have ln(x), and I want to get back x, what operation do I do to it?

EDIT: and PLEASE solve the problem entirely algebraically first. It's such a good habit to get into. Carrying this needless clutter of numbers through successive steps of the problem is just so useless. EDIT: and as I said before, it has the added benefit that you will have derived a general expression for 'r' in terms of the other quantities.

 

[Gothika]

Oh, that makes much more sense and I solved the problem. Thank you very much for taking the time to help me through it.

 

[cepheid]

Glad to be of help. There is a simpler solution method that doesn't involve natural logs that you can use to check your answer. The right-hand side (which has the 'r') has been raised to the power of "nt". What inverse operation could you carry out to get rid of this exponent on the right-hand side?