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Chapter 3: Problem 58

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

Short Answer

Expert verified
The solution for the logarithmic equation is \(x = 1.82\)

Step by step solution

01

Use the properties of logarithms

Use the properties of logarithms, specifically the quotient rule which states that \(\ln a - \ln b = \ln (a/b)\). This changes the equation into \(\ln((x+1)/(x-2)) = \ln x\).
02

Simplify the equation

As the two expressions are equal, the expressions within the logarithm must be same as well. So, write the equation as \((x+1)/(x-2) = x\).
03

Solve for x

To solve for x, first cross-multiply and distribute the x, then collect terms. It will lead to quadratic equation \(2x^2 - x - 1 = 0\). Solve this using the quadratic formula \(x = [-b \pm sqrt(b^2 - 4ac)]/(2a)\).
04

Compute the solutions

Plug in the values \(a = 2\), \(b = -1\), and \(c = -1\) into the quadratic formula, which yields \(x = 1.82\) and \(x = -0.28\).
05

Check the solutions

Check the solutions in the domain for all logarithms in the original equation. The solution \(x = -0.28\) is outside the valid domain \(x+1 > 0\), \(x-2 > 0\) and \(x > 0\), thus rejected. So, the valid solution is \(x = 1.82\) which is rounded to three decimal places.

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Most popular questions from this chapter

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