Unlocking the Power of Compound Interest- Discovering Time Without the Need for Logarithms

How to Find Time in Compound Interest Without Log

Compound interest is a powerful concept that allows your investments to grow exponentially over time. However, calculating the time it takes for an investment to reach a certain amount can be challenging, especially when using logarithms. In this article, we will explore how to find time in compound interest without relying on logarithms, providing a more accessible approach for everyone.

First, let’s understand the basic formula for compound interest:

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

Where:

• A is the future value of the investment
• P is the principal amount (initial investment)
• r is the annual interest rate (as a decimal)
• n is the number of times interest is compounded per year
• t is the number of years

Our goal is to find the value of t when A is known. To do this without using logarithms, we will rearrange the formula to solve for t:

t = log(A/P) / (n log(1 + r/n))

However, since we want to avoid logarithms, we need to find an alternative method. One approach is to use the Taylor series expansion for the exponential function. The Taylor series expansion of e^x is given by:

e^x = 1 + x + x^2/2! + x^3/3! + …

Using this expansion, we can approximate the exponential function in our compound interest formula. Let’s rewrite the formula with the Taylor series expansion:

A = P(1 + r/n)^(nt) ≈ P(1 + rt/n + (rt/n)^2/2! + (rt/n)^3/3! + …)

Now, we can set A equal to a known value and solve for t:

A = P(1 + rt/n + (rt/n)^2/2! + (rt/n)^3/3! + …)

A/P = (1 + rt/n + (rt/n)^2/2! + (rt/n)^3/3! + …)

Let’s assume A/P is approximately equal to 1 + rt/n, as higher-order terms become negligible for small values of rt/n:

1 + rt/n ≈ A/P

rt/n ≈ A/P – 1

t ≈ (A/P – 1) n / r

This approximation gives us the time it takes for the investment to grow to a certain amount without using logarithms. While this method is less accurate than using logarithms, it provides a simpler approach for those who are not comfortable with logarithmic calculations.

In conclusion, finding time in compound interest without using logarithms can be achieved by approximating the exponential function using the Taylor series expansion. This method is less precise but offers a more accessible alternative for those who prefer not to deal with logarithmic calculations.