Compound Interest Formula and Natural Logarithms

And just to be clear, the solution is:In summary, using natural logarithms, the compound interest formula can be solved for r by dividing both sides of the equation by 720 and then exponentiating both sides to get r = 360(e^(ln(1.4)/720) - 1).
  • #1
Gothika
3
0

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?
 
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  • #2
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.
 
  • #3
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
 
  • #4
Gothika said:
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.
 
  • #5
Oh, that makes much more sense and I solved the problem. Thank you very much for taking the time to help me through it.
 
  • #6
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?
 

FAQ: Compound Interest Formula and Natural Logarithms

What is the compound interest formula?

The compound interest formula is used to calculate the total amount of interest earned on a principal amount over a specific period of time. It is represented by the equation A = P(1 + r/n)^(nt), where A is the final amount, P is the principal amount, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the number of years.

How do natural logarithms relate to compound interest?

Natural logarithms are used in the compound interest formula to calculate the exponent in the equation. The natural logarithm of a number is the power to which e (the base of natural logarithms) must be raised to equal that number. In the compound interest formula, the exponent (nt) represents the number of compounding periods (n) multiplied by the number of years (t).

What is the difference between simple and compound interest?

Simple interest is calculated only on the principal amount, whereas compound interest is calculated on both the principal and the accumulated interest. This means that with simple interest, the interest earned each year remains the same, but with compound interest, the interest earned increases each year as it is added to the principal amount.

How does the frequency of compounding affect the compound interest formula?

The more frequently the interest is compounded, the higher the total amount of interest earned. This is because with more frequent compounding, the principal amount increases at a faster rate, resulting in higher interest calculations. The compound interest formula takes into account the compounding frequency by dividing the annual interest rate (r) by the number of compounding periods per year (n).

How can natural logarithms be used to calculate compound interest without a calculator?

Natural logarithms can be used to simplify the calculation of compound interest without a calculator by using the fact that ln(1+x)≈x for small values of x. This approximation can be used to simplify the compound interest formula to A≈P(1+rt), which is easier to calculate mentally. However, this approximation may not be as accurate for larger values of x, so it is recommended to use a calculator for precise calculations.

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