What is the formula for the number of periods?

n = ln(FV ÷ PV) ÷ ln(1 + r), where r is the interest rate per period.

Can I use log base 10 instead of ln?

Yes. Any logarithm works as long as you use the same base in the numerator and denominator.

If interest is added only at the end of each period, the balance does not reach the target until the next whole period.

What if the rate is zero?

The amount never grows, so there is no finite answer. The calculator asks for a non-zero rate.

Number of Periods Formula (Solve for n)

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Number of Periods (n) Calculator

The number of periods formula answers “how long will it take?” for a single sum growing at compound interest. Starting from FV = PV × (1 + r)ⁿ, take natural logarithms of both sides and solve for n: n = ln(FV ÷ PV) ÷ ln(1 + r). The answer is in the same unit as the rate: years for an annual rate, months for a monthly rate.

Formula

n = ln(FV ÷ PV)ln(1 + r)

Where

n: number of periods needed for PV to grow to FV
PV: present value (starting amount)
FV: future value (target amount)
r: interest rate per period

Calculator

Present value (PV)

Future value (FV)

Rate per period (r) (%)

Result and step-by-step solution

Number of periods

11.8957

Round up to 12 whole periods to reach at least 10,000.00.

Ratio of future to present value
10,000.005,000.00 = 2
Take natural logs
ln(2) = 0.69314718; ln(1 + 0.06) = 0.05826891
Divide
n = 0.693147180.05826891 = 11.8957

How to use the formula

The result is usually not a whole number. If interest is only credited at the end of each period, you need to round up to the next whole period to actually reach the target, and the calculator tells you that number. With continuous or daily compounding the fractional answer is meaningful as it stands.

In the example, 5,000 at 6% per period takes 11.90 periods to grow to 10,000, so 12 whole years with annual compounding. Because the target is exactly double, this is also the doubling time. For the number of payments needed to clear a loan, see the loan payoff time calculator, which includes regular payments.

Frequently asked questions

What is the formula for the number of periods?: n = ln(FV ÷ PV) ÷ ln(1 + r), where r is the interest rate per period.
Can I use log base 10 instead of ln?: Yes. Any logarithm works as long as you use the same base in the numerator and denominator.
Why round up?: If interest is added only at the end of each period, the balance does not reach the target until the next whole period.
What if the rate is zero?: The amount never grows, so there is no finite answer. The calculator asks for a non-zero rate.

Last reviewed: September 26, 2026. Calculations run in your browser and were checked against numpy-financial, SciPy and published spreadsheet examples.