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Calculus IChapter 1: ReviewSection 1-9

Exponential and Logarithm Equations — Question 8

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Question 8

Solve the equation: 5x+1=23x−25^{x+1} = 2^{3x - 2}

Instructions: Solve the equation algebraically using logarithms. Provide both an exact expression and an approximate value rounded to 3 decimal places.

Original worksheet page 1: question and worked solution for 1-9-008
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Question 8 - Solution

Taking natural logarithms (both sides are positive) gives

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

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

x=−ln⁡5−2ln⁡2ln⁡5−3ln⁡2≈6.374.\boxed{x=\frac{-\ln5-2\ln2}{\ln5-3\ln2}\approx 6.374}.

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 1-9-008

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