Exponential Growth Formula

« Back to Glossary Index

The exponential growth formula models how quantities increase rapidly over time. Exponential growth occurs when the rate of increase is proportional to the current amount, meaning the quantity grows faster and faster over time. It is a specific form of an exponential function where the base value is greater than 1. The formula is the following:

N(t) = N₀ × eʳᵗ

In this formula:

N(t) is the amount at time t
N₀ is the starting amount
e is the mathematical constant approximately equal to 2.718
r is the growth rate
t is the amount of time

For example, if $1000 is placed in a savings account that grows continuously at a 5% annual interest rate, the balance will increase exponentially over time. Using the exponential growth formula for continuous compounding, you can calculate the amount in the account after any number of years.

To calculate the amount after 10 years using continuous compounding, apply the exponential growth formula:

N(t) = N₀ × eʳᵗ

Substituting the known values:

N(10) = 1000 × e⁰⋅⁰⁵×¹⁰

First, calculate the exponent:

0.05 × 10 = 0.5

Now:

N(10) = 1000 × e⁰⋅⁵

Using the approximate value of e⁰⋅⁵ ≈ 1.6487:

N(10) ≈ 1000 × 1.6487 = 1648.72

Thus, after 10 years, the account balance would be approximately $1648.72.

The table below shows how the rate of increase grows over time with exponential growth.