ACT Preparation / Unit 12 ACT 1119 worked example, 60 to 90 seconds
Solving an Exponential Equation
The idea
An exponential equation is solved in four steps. First, isolate the exponential expression on one side. Second, check whether both sides share a base; if so, set the exponents equal. Third, if not, take the log of both sides and bring the exponent down with the power rule. Fourth, solve the resulting linear equation, and check the answer.
In practice
Worked example: 2 · 5^(x − 1) + 3 = 53. Step one: 2 · 5^(x − 1) = 50, so 5^(x − 1) = 25. Step two: 25 = 5², so x − 1 = 2, and x = 3. Check: 2 · 5² + 3 = 53. Second: 6^(2x) = 15 has no common base, so 2x log 6 = log 15, and x = log 15 / (2 log 6), about 0.76.
Where people slip
Do not take logs before isolating: log(2 · 5^(x − 1) + 3) does not split, because the log of a sum has no rule.
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