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Chapter 3: Problem 103

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\ln (x+5)=\ln (x-1)-\ln (x+1)$$

Short Answer

Expert verified
The solution for the given logarithm equation \(x \approx 0.414\) when rounded to three decimal places. The negative solution is disregarded since it would make the argument of the logarithm negative.

Step by step solution

01

Apply Logarithmic Rule

Use the logarithmic rule \(\ln a - \ln b = \ln \frac{a}{b}\) to simplify the right side of the equation. The equation becomes: \(\ln (x+5) = \ln\left(\frac{x-1}{x+1}\right)\)
02

Set the arguments of the logarithms equal to each other

If \(\ln a = \ln b\), then \(a = b\). So, \(x + 5 = \frac{x-1}{x+1}\)
03

Solve for x

To solve for \(x\), first perform cross-multiplication to get rid of the fraction to isolate \(x\) on one side of the equation. Solve this equation to find the value of x: \(x^2 + 5x = x + 4\), giving us \(x^2 + 4x - 4 = 0\). The solutions of this quadratic equation give possible values of \(x\).
04

Check Results

The solution has to be checked if the values would maintain the validity of the logarithm. A logarithm is only valid if its argument is positive. From our solutions, only the positive solution is valid as the negative would make \(\ln (x-1)\) invalid.

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