Overview
An exponential equation has the variable in an exponent. Solve it by isolating the exponential first, then take the logarithm of both sides to bring the variable down. A real solution exists when the base is positive and not 1, and the ratio of the right side to the coefficient is positive.
Formula
From a · bˣ = c we get bˣ = c/a, then x = ln(c/a) ÷ ln(b).
How it works
Isolate the exponential
Divide both sides by a to get bˣ = c/a.
Take the logarithm
Take ln of both sides: x · ln(b) = ln(c/a).
Divide by ln(b)
This gives x = ln(c/a) ÷ ln(b).
Verify
Substitute x into the original equation.
Worked examples
Example 1: 2ˣ = 8
x = ln(8) ÷ ln(2) = 3.
Example 2: 3 · 2ˣ = 24
2ˣ = 8, so x = 3.
Frequently asked questions
Why must the base be positive?
Real powers of negative numbers are not defined for all real exponents.
What if c/a is negative?
There is no real solution, because bˣ is always positive when b > 0.
Can I use base e?
Yes, and the natural logarithm simplifies it directly since ln(eˣ) = x.
This calculator is for educational and informational purposes only and is not a substitute for professional advice.