Exponential Equation Solver (a·bˣ = c)

Understanding the exponential equation solver — concept and principle

An exponential equation has the unknown in the exponent. This solver handles the common form a · bˣ = c and returns the exact value of x using logarithms. Equations of this type appear whenever something grows or decays by a constant factor: compound interest, population growth, radioactive decay, doubling time, half-life and cooling problems.

The exponent x simply counts how many times 5 must double to reach 160.

How it is calculated

Divide both sides by a, then take the logarithm of both sides:

bˣ = c ÷ a
x · ln b = ln(c ÷ a)
x = ln(c ÷ a) ÷ ln b

Here a is the starting value, b is the growth factor per step (b > 1 grows, 0 < b < 1 decays) and c is the target value. x is the number of steps, in whatever unit one step represents (years, periods, half-lives).

The principle

The logarithm is the inverse of exponentiation: ln b asks "what power of e gives b?". Its key property ln(bˣ) = x · ln b turns an unknown exponent into an ordinary multiplication, which is easy to solve. Any logarithm base works, because the ratio ln(c/a) ÷ ln b equals log(c/a) ÷ log b in base 10 or any other base. This is the change-of-base formula: x = log_b(c ÷ a).

Worked example

Default inputs: a = 5, b = 2, c = 160, so the equation is 5 · 2ˣ = 160.
Divide by 5: 2ˣ = 32.
Take logarithms: x = ln 32 ÷ ln 2 = 3.4657359 ÷ 0.6931472.
Result: x = 5. Check: 5 · 2⁵ = 5 · 32 = 160.

Each point is the calculator re-run with only “a” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

a	x
1.25	7
2.5	6
3.75	5.415037499
5	5
6.25	4.678071905
7.5	4.415037499
10	4

How to read the result

A positive x means c is reached after x steps forward; a negative x means c lay in the past relative to a (for growth, c is smaller than a). A non-integer x is perfectly normal: x = 4.3 means 4.3 periods. Because x = log_b(c ÷ a), doubling the target c always adds the same amount, ln 2 ÷ ln b, to x; with b = 2 that is exactly one more step.

Common mistakes

Taking the logarithm before dividing by a: ln(5 · 2ˣ) is ln 5 + x ln 2, not x ln 10.
Entering a percentage rate as b. A 7% growth rate means b = 1.07, not 7 or 0.07.
Using b = 1 or b ≤ 0, for which the equation has no meaningful solution.
Choosing c and a with opposite signs; c ÷ a must be positive.

Keep in mind: The solver requires c ÷ a > 0, b > 0 and b ≠ 1. Outside these conditions there is no real solution and the result will be undefined or meaningless.

Frequently asked questions

How do I solve an equation like 3ˣ = 2ˣ⁺¹?

Take logs of both sides: x ln 3 = (x + 1) ln 2, so x = ln 2 ÷ (ln 3 − ln 2) ≈ 1.7095.

How long does money take to double at 5% per year?

Solve 1.05ˣ = 2: x = ln 2 ÷ ln 1.05 ≈ 14.2 years (ignoring tax and fees).

Is the "rule of 72" accurate?

It is an approximation: 72 ÷ rate in percent. At 5% it gives 14.4 years against the exact 14.2.

Sources: NIST Digital Library of Mathematical Functions, §4.2 Logarithm, Exponential, Powers (dlmf.nist.gov) · Stewart, J., Calculus: Early Transcendentals, §1.5 Inverse Functions and Logarithms

Checked 8 Nov 2026. Standard formulas, computed in your browser with double-precision numbers.

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