Solve each equation. Give solutions in exact form. ln(4x - 2) - ln 4 = -ln(x - 2)

Verified step by step guidance

1. Recall the logarithmic property that allows you to combine the difference of logarithms: \(\ln a - \ln b = \ln \left( \frac{a}{b} \right)\). Apply this to the left side of the equation to combine the logarithms: \(\ln(4x - 2) - \ln 4 = \ln \left( \frac{4x - 2}{4} \right)\).
2. Rewrite the equation using the combined logarithm: \(\ln \left( \frac{4x - 2}{4} \right) = -\ln(x - 2)\).
3. Use the property that \(-\ln y = \ln \left( \frac{1}{y} \right)\) to rewrite the right side: \(\ln \left( \frac{4x - 2}{4} \right) = \ln \left( \frac{1}{x - 2} \right)\).
4. Since the natural logarithm function \(\ln\) is one-to-one, set the arguments equal to each other: \(\frac{4x - 2}{4} = \frac{1}{x - 2}\).
5. Solve the resulting rational equation for \(x\) by cross-multiplying and simplifying. Remember to check for any restrictions on \(x\) from the original logarithmic expressions to ensure the solutions are valid.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms, such as the difference rule ln(a) - ln(b) = ln(a/b), is essential for simplifying and combining logarithmic expressions. These properties allow you to rewrite the equation in a more manageable form to isolate the variable.

Solving Logarithmic Equations

Solving logarithmic equations involves rewriting the equation to isolate the logarithm and then exponentiating both sides to eliminate the logarithm. This process converts the equation into an algebraic form that can be solved for the variable.

Domain Restrictions of Logarithmic Functions

Logarithmic functions are only defined for positive arguments. When solving equations involving logarithms, it is crucial to consider domain restrictions to ensure that the solutions do not make any logarithmic expression undefined or invalid.