Calxsolver
Precalculus

Exponential Equation Solver

An exponential equation has the unknown in the exponent. To solve a·bˣ = c, this tool isolates the power, then applies logarithms to bring the exponent down: x = ln(c/a) / ln(b).

Exponential Equation Solver

Solve a·bˣ = c for the exponent x.

Try:
Answerx = 5
  1. Equation1·2^x = 32
  2. Isolate the power2^x = c / a = 32
  3. Take logarithmsx = ln(32) / ln(2)
  4. Solvex = 5
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Formula and method

a·bˣ = c → x = ln(c/a) / ln(b) Requires: b > 0, b ≠ 1, c/a > 0

Divides both sides by a to isolate the exponential term bˣ = c/a, then applies the natural logarithm to both sides: x·ln(b) = ln(c/a), giving x = ln(c/a) / ln(b). Returns no solution when c/a ≤ 0 (a positive base raised to any real power is always positive) or when the base b ≤ 0 or b = 1.

Worked examples

Solve 2ˣ = 32 Use logarithms to isolate the unknown exponent. Solve 3ˣ = 27 An exponential equation that resolves to a whole-number exponent.

Learn why this works

Why eˣ Differentiates to Itself Why d/dx eˣ = eˣ and the role of e in calculus →

Key terms

Logarithm The inverse of exponentiation: logₐ(x) is the exponent to which a must be raised to get x.

Frequently asked questions

Why do we use logarithms?

Logarithms are the inverse of exponentiation, so they move the unknown out of the exponent and into a solvable expression.

When is there no solution?

If c/a is zero or negative there is no real solution, because a positive base raised to any real power stays positive.

What values can the base take?

The base b must be positive and not equal to 1; b = 1 would make the left side constant.

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