Calculator Inputs
Enter two exponential equations in this format: A · b(px + qy) = C. The solver converts both equations into a linear system using logarithms.
Plotly Graph
The chart below shows the logarithmically transformed system. Each exponential equation becomes a straight line in the x-y plane.
Example Data Table
| Example | A1 | b1 | p1 | q1 | C1 | A2 | b2 | p2 | q2 | C2 | Expected x | Expected y |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reference case | 2 | 3 | 1 | 1 | 54 | 5 | 2.718281828 | 2 | -1 | 5 | 1 | 2 |
| Meaning | 2 × 3^(x + y) = 54 | 5 × e^(2x − y) = 5 | Unique intersection | |||||||||
Formula Used
Equation 1: A1 · b1(p1x + q1y) = C1
Equation 2: A2 · b2(p2x + q2y) = C2
Take natural logarithms:
ln(C1/A1) = (p1x + q1y) ln(b1)
ln(C2/A2) = (p2x + q2y) ln(b2)
Linearized system:
α1x + β1y = r1
α2x + β2y = r2
where α = p ln(b), β = q ln(b), and r = ln(C/A).
Determinant test:
D = α1β2 − α2β1. If D ≠ 0, then a unique solution exists.
Closed-form solution:
x = (r1β2 − r2β1) / D
y = (α1r2 − α2r1) / D
How to Use This Calculator
- Enter A, base, x coefficient, y coefficient, and right-side constant for both equations.
- Keep each base positive and different from 1.
- Ensure C/A stays positive for both equations.
- Set graph ranges and decimal precision for better viewing.
- Click Solve System to compute x and y.
- Review determinant, transformed equations, and residual values.
- Use CSV or PDF export to save your results.
Frequently Asked Questions
1) What kind of exponential systems can this solver handle?
This solver handles two equations shaped like A · b^(px + qy) = C. Each equation may use different coefficients, constants, and bases.
2) Why must each base be positive and not equal to 1?
Logarithmic transformation requires positive bases. A base of 1 removes exponential behavior and makes the transformed coefficients collapse to zero.
3) Why does the calculator use logarithms?
Logarithms convert exponential expressions into linear equations. That makes the system easier to solve with determinant methods and intersection graphs.
4) What does a zero determinant mean here?
A zero determinant means the transformed lines are either dependent or inconsistent. The system then has infinitely many solutions or no unique real solution.
5) Can I use negative values for A or C?
Yes, but the ratio C/A must stay positive. Otherwise the logarithm is undefined and the real-valued solution process breaks.
6) What are residuals?
Residuals measure how closely the computed solution satisfies the original equations. Smaller residuals mean a more accurate numerical fit.
7) Why do the plotted curves look like straight lines?
The graph shows the transformed system, not the raw exponential equations. After applying logarithms, each equation becomes linear in x and y.
8) When should I change graph range and precision?
Use wider graph ranges when the intersection falls outside the default view. Increase precision when coefficients or residuals are very small.