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Chapter 3: Problem 55

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\ln x-\ln (x+1)=2$$

Short Answer

Expert verified
The solution for \(x\) is \(e^2 / (e^2 - 1)\). After three decimal approximation, \(x 鈮� 7.389 / 6.389 鈮� 1.157\)

Step by step solution

01

Simplified Equation

Use the logarithmic property \(\ln a - \ln b = \ln(a/b)\) to simplify the given equation: \( \ln x - \ln (x+1) = 2\) becomes \( \ln (x / (x+1)) = 2\)
02

Convert to Exponential Form

This equation can now be transformed from logarithmic form to exponential form. Base e and exponent 2 is equal to the argument of the logarithm: \(\ln (x / (x+1)) = 2\) becomes \(e^2 = x / (x+1)\)
03

Working Out the Equation

Now multiply through by \(x + 1\) to remove the fraction: \(e^2 \cdot (x + 1) = x\). This simplification leads to \(e^2x + e^2 = x\)
04

Solving for x

Rearrange this equation by subtracting \(e^2x\) and \(x\) on both sides to isolate x: \(e^2x - x = e^2\)
05

Factoring x

Next, factor out \(x\) on the left hand side to get: \(x(e^2 - 1) = e^2\)
06

Final Solution

Finally, solving for \(x\), divide through by \((e^2 - 1)\) to get: \(x = e^2 / (e^2 - 1)\)

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