Exponential growth: formula, examples and how to measure it

What is exponential growth?

Exponential growth happens when a quantity grows by the same percentage in every period, so each step adds more than the one before. Linear growth adds the same amount each period; exponential growth multiplies by the same factor.

A user base that grows 8% a week adds 160 users in the first week from 2,000, but about 400 a week by week twelve, because 8% of a bigger number is a bigger number.

The exponential growth formula

For growth measured per period (per day, week, month):

N(t) = N₀ × (1 + r)ᵗ

• N₀ is the starting value
• r is the growth rate per period, as a decimal (8% = 0.08)
• t is the number of periods

For continuous growth, the same idea is written with e:

N(t) = N₀ × eᵏᵗ,   where k = ln(1 + r)

An 8% weekly rate is a continuous rate of k = ln(1.08) ≈ 0.077. For business metrics the discrete form is easier to read; the continuous form is common in science and finance.

Worked example: exponential vs linear

A product has 2,000 weekly active users (sample data). Compare two futures over 12 weeks:

For the first month the two lines look almost the same. By week 12 the exponential path is 28% higher, and the gap keeps widening. That early resemblance is why exponential trends are easy to underestimate.

Doubling time and the rule of 70

The time it takes an exponentially growing quantity to double is:

doubling time = ln(2) ÷ ln(1 + r)

At 8% a week that is 9.0 weeks. A quick mental shortcut is the rule of 70 (or 72): divide 70 by the percentage rate. 70 ÷ 8 ≈ 8.8 weeks; 72 ÷ 8 = 9 weeks. The shortcut is accurate for rates under about 10% per period.

How to measure the growth rate from your data

If a metric went from N₀ to N over t periods, the average per-period growth rate is:

r = (N ÷ N₀)^(1/t) − 1

Weekly actives rose from 2,000 to 3,200 in six weeks: (3,200 ÷ 2,000)^(1/6) − 1 = 8.1% per week. This is the same calculation as the compound monthly growth rate applied to weeks.

Two checks before you call something exponential:

• Plot it on a log scale. Exponential growth is a straight line on a logarithmic y-axis. If the line bends downward, growth is slowing.
• Look at the period-over-period percentages, not the absolute changes. Steady percentages mean exponential; steady absolute changes mean linear. See month-over-month and year-over-year.

Why exponential growth stops: logistic growth

Real growth runs into limits: the number of potential customers, budget, server capacity, word of mouth reaching everyone in a niche. The logistic model adds a ceiling K:

N(t) = K ÷ (1 + ((K − N₀) ÷ N₀) × e^(−kt))

Same starting point and rate as above, with a ceiling of 10,000 users:

Early on, logistic growth looks exponential. It then slows and flattens near the ceiling, the S-curve you also see in the technology adoption curve. Projecting an early exponential trend a year ahead is how plans end up an order of magnitude too optimistic.

Exponential decay

The same math with a negative rate describes decline. A cohort of 1,000 subscribers losing 10% a month keeps 1,000 × 0.9¹² ≈ 282 after a year, and halves in about 6.6 months. Constant-percentage churn is exponential decay, which is why small improvements in monthly churn compound into large differences in annual recurring revenue.

Common mistakes

• Adding percentages instead of compounding. 10% a month is not 120% a year; it is 1.1¹² − 1 = 214%.
• Mixing percentage points with percent. A conversion rate going from 2% to 3% is +1 percentage point and +50%.
• Calling any fast growth exponential. Viral loops need a viral coefficient and cycle time that hold steady; most products see a burst, then a slowdown.
• Extrapolating from a short window. Fit the rate over several periods, and check it against seasonality.

In TapCub

Growth reports show active users, sign-ups and revenue over time, and every chart can compare the current range with the previous period or the same period last year. To check whether a trend is exponential, compare period-over-period percentages, or export the series and plot it on a log scale. Goals and growth stages live under Insights; see growth loop for the mechanics that can produce sustained compounding.

Sources

• OpenStax. College Algebra 2e, section 6.7: Exponential and Logarithmic Models.
• Khan Academy. Exponential growth and decay.
• Wikipedia. Rule of 72 (also covers the rule of 70 and 69.3).