Exponential Equations Solver
Enter any exponential equation and get the full step-by-step solution — same-base matching, taking logarithms, or substitution when two exponential terms appear.
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Matching bases in an exponential equation — worked step by step
Below is one fully worked example plus a short primer so you can see exactly how our AI reasons through a problem.
Example Problem
DEMO- 1
Rewrite both sides as powers of 2
4 = 2² and 8 = 2³. Converting to a common base lets us compare exponents directly.
- 2
Apply the power-of-a-power rule on the left
Multiply the outer exponent into the bracket: 2(x − 1).
- 3
Set exponents equal — bases are the same
If aᵐ = aⁿ and a ≠ 1, then m = n.
- 4
Solve the linear equation
Add 2 to both sides, then divide by 2.
- 5
Verify in the original equation
4^(5/2 − 1) = 4^(3/2) = (√4)³ = 2³ = 8 ✓
Final Answer
How to solve exponential equations — three strategies for every case
An exponential equation is any equation where the unknown appears in an exponent. Two core strategies cover most problems. When both sides can be written as powers of the same base, set the exponents equal and solve the resulting linear or polynomial equation — this matching-bases technique avoids logarithms entirely. When the bases cannot be matched (for example 3^x = 7), take logarithms of both sides and apply the log power rule to bring the exponent down to a coefficient: x = log 7 / log 3.
A third family uses substitution: an equation like 4^x − 6·2^x + 8 = 0 becomes a quadratic in u after substituting u = 2^x. Solve for u, then recover x by taking the logarithm. Exponential equations appear across Class 11/12 CBSE, JEE Advanced, A-Level Maths, and AP Precalculus. Recognising which strategy fits — same base, log method or substitution — is the key skill examiners test.
Exponential equations to practise (Class 11, JEE, A-Level)
Tap any problem to solve it with full step-by-step working.
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1. 3^{2x} = 81
Matching basesClass 11 / GCSEEasy - Solve with AI →
2. 5^{x+2} = 125
Matching bases — shifted exponentClass 11Easy - Solve with AI →3.Logarithm methodClass 12 / A-LevelMedium
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4. e^{2x-1} = e^{x+3}
Natural exponential — matching exponentsA-Level / AP PrecalculusEasy - Solve with AI →5.Quadratic substitution (u = 3^x)Class 12 / JEE / A-LevelHard
Frequently asked questions
What is an exponential equation?+
An exponential equation is one where the variable appears in the exponent, such as 2^x = 32 or 3^(2x+1) = 9. The standard approach is to express both sides with the same base and set exponents equal, or to take logarithms when a common base doesn't exist.
How do I solve an exponential equation by matching bases?+
Write both sides as powers of the same base, then set the exponents equal and solve. For example, 8^x = 32 becomes 2^(3x) = 2^5, so 3x = 5 and x = 5/3. This works whenever both numbers are exact powers of a common base — powers of 2, 3, 5 or e are the most common cases.
How do I solve an exponential equation using logarithms?+
Take the log (base 10 or natural log) of both sides, then apply the power rule: log(aˣ) = x·log(a). For 3^x = 7, this gives x·log 3 = log 7, so x = log 7 / log 3 ≈ 1.771. Any consistent log base works — log₁₀ and ln are both fine.
What is the substitution method for exponential equations?+
When an equation has two exponential terms with the same base — like 9^x − 4·3^x + 3 = 0 — let u equal the smaller power (u = 3^x). Since 9^x = (3^x)² = u², the equation becomes u² − 4u + 3 = 0, a factorable quadratic. Solve for u, then solve 3^x = u using logarithms.
Are exponential equations in CBSE Class 11/12 and JEE?+
Yes. CBSE Class 11/12 covers exponential equations under the logarithm and sequences chapters. JEE Mains and Advanced include them in algebra, often combined with inequalities or the substitution technique. A-Level Maths and AP Precalculus also include exponential and logarithmic equations as a core topic.
What is the difference between an exponential equation and an exponential function?+
An exponential function is a general expression like f(x) = aˣ that describes growth or decay. An exponential equation is a specific statement where two expressions involving exponents are set equal — the goal is to find the value(s) of x that satisfy it. Solving an exponential equation typically uses either matching bases or logarithms.
Is there a free exponential equations solver with step-by-step working?+
Yes — this solver shows every step from identifying the strategy to the final value of x, including the log calculation when needed. The free plan gives 5 daily solves with no credit card required.
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