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Calculus IChapter 1: ReviewSection 1-9

Exponential and Logarithm Equations — Question 1

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Question 1

Solve the following equation for xx: 32x−1=7x+23^{2x - 1} = 7^{x + 2}

Instructions: Solve algebraically using logarithms and express your answer in exact form. Approximate the result to 3 decimal places.

Original worksheet page 1: question and worked solution for 1-9-001
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Question 1 - Solution

Taking natural logarithms (both sides are positive) gives

(2x−1)ln⁡3=(x+2)ln⁡7.(2x-1)\ln3=(x+2)\ln7.

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

x=ln⁡3+2ln⁡72ln⁡3−ln⁡7≈19.857.\boxed{x=\frac{\ln3+2\ln7}{2\ln3-\ln7}\approx 19.857}.

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 1-9-001

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