Exponential equations
Objective. Solve equations by common bases or logarithms.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Exponential equations have the unknown in the power, and are solved either by writing both sides with a common base or by taking logarithms of both sides. If the bases match, say 2ˣ = 2⁵, then exponents can be equated directly to give x = 5. When bases cannot be matched, take logs of both sides: for 3ˣ = 20, taking ln gives x×ln(3) = ln(20), so x = ln(20)/ln(3). For example 5ˣ = 125 becomes 5ˣ = 5³ by rewriting 125, giving x = 3 instantly. Choosing a common base is quicker when possible, but logarithms always work, even for non-integer or messy exponents.
Worked example
Solve the exponential equation 4ˣ = 9, giving your answer to 3 significant figures.
- 1Take natural logs of both sides: ln(4ˣ) = ln(9).
- 2Apply the power law: x×ln(4) = ln(9).
- 3Divide: x = ln(9)/ln(4) = 2.1972.../1.3863...
- 4Compute and round: x ≈ 1.58.
Answer: x ≈ 1.58
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Practice
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Start practisingCommon mistake. A frequent mistake is dividing both sides by the base instead of taking logs, turning 3ˣ = 20 into x = 20/3. The variable is in the exponent, so logarithms, not division, must be used to bring it down.
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