Problem 91 - College Algebra Essentials

Question

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\ln (x-2)-\ln (x+3)=\ln (x-1)-\ln (x+7)$$

Step-by-Step Solution

Answer

\(x = \frac{11}{3}\) or approximately 3.67

Step 1: Combine the Logarithms

Using the properties of logarithms, we can rewrite the equation as: \(\ln \frac{x-2}{x+3} = \ln\frac{x-1}{x+7}\)

Step 2: Isolate the Variables

Setting the arguments of the two natural logarithms equal to each other, we get: \(\frac{x-2}{x+3} = \frac{x-1}{x+7}\)

Step 3: Solve for \(x\)

Clearing off the denominators and solving for \(x\), \((x-2)(x+7) = (x+3)(x-1)\) which simplifies to \(x^2+5x-14 = x^2+2x-3\). Subtracting \(x^2\), \(2x\), and \(-3\) from both sides gives: \(3x-11 = 0\). Finally, solving for \(x\) gives: \(x = \frac{11}{3}\)

Step 4: Check for Validity

The values which would make each logarithm undefined are those that make the arguments less than or equal to 0. Hence, checking for validity of \(x = \frac{11}{3}\), we find that it makes none of the arguments of the logs in the original equation less than or equal to zero. Therefore, \(x = \frac{11}{3}\) is valid.

Step 5: Decimal Approximation

The decimal value of \(\frac{11}{3}\) is about 3.67, rounded to two decimal places.