Solving Exponential Equations

This lesson focuses on the techniques for solving equations where the unknown appears in the exponent. These questions feature heavily in A-Level papers and draw on your knowledge of index laws and logarithm laws.

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Type 1: Equations Reducible to a Common Base

If both sides can be written as powers of the same base, equate the indices.

Example 1: Solve 4ˣ = 32

• Write both sides as powers of 2: (2²)ˣ = 2⁵
• 2²ˣ = 2⁵
• 2x = 5
• x = 5/2

Example 2: Solve 9^(2x-1) = 27ˣ

• Write as powers of 3: (3²)^(2x-1) = (3³)ˣ
• 3^(4x-2) = 3^(3x)
• 4x - 2 = 3x
• x = 2

Exam Tip: Always look for a common base first. This avoids logarithms entirely and gives exact answers.

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Type 2: Taking Logarithms of Both Sides

When the bases cannot be matched, take logs (base 10 or natural log) of both sides.

Example: Solve 3ˣ = 7

• Take log₁₀ of both sides: log(3ˣ) = log(7)
• x × log(3) = log(7) [Power Law]
• x = log(7) / log(3) = 0.8451.../0.4771... ≈ 1.771