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Calculus IChapter 1: ReviewSection 1-9

Exponential and Logarithm Equations — Question 4

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Question 4

Solve for xx in the equation: 23x+1=5x−22^{3x + 1} = 5^{x - 2}

Give your answer in exact form using logarithms, and approximate the result to three decimal places.

Original worksheet page 1: question and worked solution for 1-9-004
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Question 4 - Solution

Taking natural logarithms (both sides are positive) gives

(3x+1)ln⁡2=(x−2)ln⁡5.(3x+1)\ln2=(x-2)\ln5.

Collect the terms in xx and divide by their nonzero coefficient:

x=−ln⁡2−2ln⁡53ln⁡2−ln⁡5≈−8.323.\boxed{x=\frac{-\ln2-2\ln5}{3\ln2-\ln5}\approx -8.323}.

The logarithm is one-to-one, so solving this equivalent linear equation introduces no extraneous solutions. The exact expression verifies the original equation.

Original worksheet page 2: question and worked solution for 1-9-004

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