Exponential equations

Objective. Rewrite with common bases to solve equations in indices.

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Learn: the key idea

Exponential equations have the unknown in the power, and are solved by rewriting every term with the same base so the exponents themselves can be equated, since a^p = a^q means p = q when a>0 and a≠1. Begin by expressing numbers like 4, 8, 16 or 1/9 as powers of a common base such as 2 or 3. Once both sides share the base, drop the base and solve the resulting linear or quadratic equation in the exponent. For example, 9^x = 27 becomes (3²)^x = 3³, so 2x=3 and x=3/2.

Worked example

Solve the exponential equation 25^(x+1) = 125^x.

1 $Write both sides as powers of 5: 25 = 5², 125 = 5³$
2 $Rewrite the equation: 5^(2(x+1)) = 5^(3x)$
3 $Equate exponents: 2x+2 = 3x$
4 $Solve for x: 2 = 3x - 2x, so x = 2$

Answer: x = 2

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Common mistake. Pupils often take logarithms unnecessarily or divide the bases instead of finding a common base, turning 2^x = 8 into x = 8÷2 = 4; instead rewrite 8 as 2³ and equate exponents to get x=3.

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