How do you solve an exponential equation that is quadratic in form, like e^(2x) − 5e^x + 6 = 0?

Substitute u = e^x to get u² − 5u + 6 = 0. Factor or use the quadratic formula: (u − 2)(u − 3) = 0, so u = 2 or u = 3. Back-substitute: e^x = 2 → x = ln(2) ≈ 0.693147, and e^x = 3 → x = ln(3) ≈ 1.098612. Only positive values of u are valid since e^x > 0 always.

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Exponential Equation Solver

aˣ = b · Linear Exponent · Natural e · Quadratic in eˣ

Solve exponential equations step by step using logarithms. Supports a^x = b, a^(mx+n) = b, natural log forms, and quadratic-in-eˣ equations. Exact + decimal answers with graph.

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Laws of Logarithms — Quick Reference

Product Rule

log(AB) = log A + log B

Quotient Rule

log(A/B) = log A − log B

Power Rule

log(Aⁿ) = n · log A

Change of Base

logₐ(b) = log(b)/log(a)

Natural Log Identity

ln(eˣ) = x

Log of 1

logₐ(1) = 0 (any a)

Log of Base

logₐ(a) = 1

Inverse Property

a^(logₐ(x)) = x

What Are Exponential Equations?

An exponential equation is any equation in which the unknown variable appears in an exponent. The simplest form is aˣ = b, asking "to what power must we raise a to get b?" These equations arise constantly in science, finance, and engineering whenever quantities grow or decay multiplicatively over time.

Unlike polynomial equations, exponential equations cannot be solved by standard algebraic manipulation alone. The key tool is the logarithm: since log(aˣ) = x·log(a), taking the log of both sides converts an exponential equation into a linear one that is easy to solve.

How to Solve Exponential Equations Step by Step

Type 1 — Basic Form: aˣ = b

Take the logarithm (any base) of both sides and apply the power rule:

Take log: log(aˣ) = log(b)
Power rule: x · log(a) = log(b)
Isolate x: x = log(b) / log(a) (change-of-base formula)

Example: 5ˣ = 100 → x = log(100)/log(5) = 2/log(5) ≈ 2.861353

Type 2 — Linear Exponent: a^(mx+n) = b

Take log of both sides, expand, then solve the resulting linear equation:

Take log: (mx + n) · log(a) = log(b)
Divide: mx + n = log(b)/log(a)
Subtract n: mx = log(b)/log(a) − n
Divide by m: x = (log(b)/log(a) − n) / m

Type 3 — Coefficient Form: k·aˣ = b

First divide both sides by k to isolate the exponential term: aˣ = b/k. Then proceed with the basic form. Note: if b/k ≤ 0 there is no real solution.

Type 4 — Natural Exponential: eˣ = c or e^(ax+b) = c

Apply the natural logarithm (ln) since ln and e are inverse operations:

eˣ = c → x = ln(c)
e^(ax+b) = c → ax + b = ln(c) → x = (ln(c) − b) / a

Type 5 — Quadratic in eˣ

Equations like e^(2x) − 5eˣ + 6 = 0 become quadratics after substitution u = eˣ:

Substitute: u² − 5u + 6 = 0
Factor or quadratic formula: u = 2 or u = 3
Back-substitute: x = ln(2) ≈ 0.693147 or x = ln(3) ≈ 1.098612
Discard any u ≤ 0 since eˣ > 0 always

Where Exponential Equations Appear in Real Life

Field	Equation	What x represents
Compound interest	A = P · e^(rt)	Time to reach amount A
Radioactive decay	N = N₀ · e^(−λt)	Time until quantity N
Population growth	P = P₀ · a^t	Time to reach population P
pH chemistry	pH = −log[H⁺]	H⁺ concentration
Earthquake (Richter)	M = log(A/A₀)	Amplitude ratio
Sound (decibels)	dB = 10 · log(I/I₀)	Intensity ratio

Natural Log vs. Common Log

Both the natural logarithm (ln, base e) and the common logarithm (log, base 10) produce the same final answer for x because of the change-of-base formula: ln(b)/ln(a) = log(b)/log(a). Use ln when the base is e (it gives the cleanest form), and log for all other bases. Most scientific calculators provide both.

Common Mistakes to Avoid

Forgetting that a must be positive and not equal to 1 for aˣ = b to have a unique solution.
Taking log of a negative number — since aˣ > 0 always, if b ≤ 0 there is no real solution.
Dropping the coefficient k in k·aˣ = b without first isolating aˣ.
Forgetting to discard negative u roots in quadratic-form equations where u = eˣ.
Using log instead of ln inconsistently — always use the same log throughout one calculation.
Rounding too early — carry full precision through all steps and round only the final answer.

Frequently Asked Questions

How do you solve an exponential equation like 2^x = 32?

Take log of both sides: log(2ˣ) = log(32), giving x·log(2) = log(32). Divide both sides by log(2): x = log(32)/log(2) = 5. You can use any base — log, ln, or log₂ — and get the same answer.

What is the change-of-base formula and why is it needed?

The change-of-base formula is logₐ(b) = log(b) / log(a) = ln(b) / ln(a). Most calculators only provide log base 10 and natural log. To evaluate log₅(100), you compute log(100)/log(5) ≈ 2.861. This formula is essential for solving aˣ = b where a is not 10 or e.

How do you solve a^(mx + n) = b for x?

Take log of both sides: (mx + n)·log(a) = log(b). Divide by log(a): mx + n = log(b)/log(a). Subtract n: mx = log(b)/log(a) − n. Divide by m: x = (log(b)/log(a) − n) / m. Each step uses basic logarithm and linear algebra properties.

How do you solve a quadratic-form equation like e^(2x) − 5e^x + 6 = 0?

Substitute u = eˣ to get u² − 5u + 6 = 0. Factor: (u − 2)(u − 3) = 0, so u = 2 or u = 3. Back-substitute: eˣ = 2 → x = ln(2) ≈ 0.693147, and eˣ = 3 → x = ln(3) ≈ 1.098612. Discard any u ≤ 0 since eˣ is always positive.

What is the difference between natural log and common log when solving exponential equations?

Natural log (ln) uses base e ≈ 2.71828; common log (log) uses base 10. For equations with base e, ln gives the cleanest form: eˣ = c → x = ln(c). For other bases, both work equally via change-of-base: x = ln(b)/ln(a) = log(b)/log(a). The numerical answer is identical.

Can an exponential equation have no solution?

Yes. Since aˣ > 0 for all real x (when a > 0, a ≠ 1), equations like 2ˣ = −5 or k·aˣ = b where b/k ≤ 0 have no real solution. In quadratic-form equations, roots with u ≤ 0 must also be discarded since eˣ is always strictly positive.

Where do exponential equations appear in real life?

Exponential equations arise in compound interest (A = P·e^(rt)), radioactive decay (N = N₀·e^(−λt)), population growth, pH calculations (pH = −log[H⁺]), the Richter scale for earthquake magnitude, sound intensity in decibels, and pharmacokinetics (drug concentration over time).