Problem 92 - 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. $$\ln (x-5)-\ln (x+4)=\ln (x-1)-\ln (x+2)$$

Step-by-Step Solution

Answer

The equation does not have a solution, as the value of \(x\) obtained does not satisfy the domain requirements of the original logarithmic expressions.

Step 1: Use the properties of logarithms

Use the properties of logarithms to condense the logarithms in the expression:\[\ln \left(\frac{x-5}{x+4}\right)=\ln \left(\frac{x-1}{x+2}\right)\]

Step 2: Simplify the equation

As the logarithms are equal, we can set the arguments equal to each other, to simplify the expression:\[\frac{x-5}{x+4}=\frac{x-1}{x+2}\]

Step 3: Solve for \(x\)

Cross multiply and solve for \(x\):\[(x-5)(x+2)=(x-1)(x+4)\]Expanding gives us:\[x^2 - 3x - 10 = x^2 -x -4\]Solving this equation:\[-2x = 6 \ x = -3\]

Step 4: Check the domain

Checking our solution against the domain restrictions set by the original logarithmic expressions we can see that \(x = -3\) does not satisfy the original domain requirements of \(x > 5\) or \(x > 1\) (since logarithms do not accept negative arguments here). Therefore, \(x = -3\) is not an acceptable solution.