Exponential Equation Solver
Solve exponential equations with different bases using logarithm properties. View step-by-step expansion and isolation of the variable x.
About
Exponential equations appear frequently in population dynamics, finance, and radioactive decay models. Solving them requires manipulating exponents that contain variables. Accuracy is paramount when calculating compound interest or bacterial growth rates. A small error in the exponent results in massive deviations in the final value. This tool handles cases where bases differ on either side of the equation. It applies logarithms to bring the variable down from the exponent. The solver isolates x by expanding terms and grouping linear components.
Formulas
The core principle relies on the property of logarithms where exponents become multipliers.
Given an equation:
b1f(x) = b2g(x)
Apply natural log (ln) to both sides:
f(x) ⋅ ln(b1) = g(x) ⋅ ln(b2)
This transforms a transcendental equation into a linear algebraic equation that is solvable for x.
Reference Data
| Base b | ln(b) Approx | log10(b) Approx | Common Power (b2) | Common Power (b3) |
|---|---|---|---|---|
| 2 | 0.693147 | 0.301030 | 4 | 8 |
| 3 | 1.098612 | 0.477121 | 9 | 27 |
| 4 | 1.386294 | 0.602060 | 16 | 64 |
| 5 | 1.609438 | 0.698970 | 25 | 125 |
| e | 1.000000 | 0.434294 | 7.389 | 20.085 |
| 6 | 1.791759 | 0.778151 | 36 | 216 |
| 7 | 1.945910 | 0.845098 | 49 | 343 |
| 8 | 2.079442 | 0.903090 | 64 | 512 |
| 9 | 2.197225 | 0.954243 | 81 | 729 |
| 10 | 2.302585 | 1.000000 | 100 | 1000 |