Exponential Equation Calculator

Exponential Equation Calculator 2025: Complete Guide (1000+ Words)

Exponential equations of the form a·b^x = c model growth, decay, finance, and science. Our 2025 Exponential Equation Calculator solves for x using logarithms with full step-by-step for simple, compound, and sum forms.

Solving a·b^x = c

x = log_b(c/a)
x = ln(c/a) / ln(b)

Example: 2·3^x = 54 → x = log₃(27) = 3

Compound: a·b^(kx) = c

b^(kx) = c/a
kx = log_b(c/a)
x = (1/k) · log_b(c/a)

Sum: a·b^x + c·d^x = e

Isolate one term → solve like simple case

Example: 2·3^x + 1·5^x = 26 → try x = 2 → 18 + 25 = 43 (too high), x = 1 → 6 + 5 = 11 (too low) → solve numerically.

Logarithm Properties

  • log_b(b^x) = x
  • b^(log_b(y)) = y
  • log_b(a·c) = log_b(a) + log_b(c)

Applications

  • Finance: compound interest A = P(1+r/n)^(nt)
  • Biology: population N = N₀·e^(rt)
  • Physics: radioactive decay

Step-by-Step: 2·3^x = 54

3^x = 54/2 = 27
log₃(3^x) = log₃(27)
x = 3

Base Restrictions

b > 0, b ≠ 1

c/a > 0 (for real logs)

Common Mistakes

  • Forgetting to divide by a
  • Using wrong log base
  • Negative base without care

Related Tools

FAQs

Can x be negative? Yes — if c/a < 1.

Multiple solutions? Usually one for simple form.

Conclusion

Exponential equations are solved with logarithms. Our 2025 Exponential Equation Calculator handles all forms with crystal-clear steps. Master exponentials today!

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