Solve each equation. Give solutions in exact form. See Examples 5–9. ln(4x - 2) - ln 4 = -ln(x - 2)

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is essential for solving logarithmic equations. Key properties include the product rule (ln(a) + ln(b) = ln(ab)), the quotient rule (ln(a) - ln(b) = ln(a/b)), and the power rule (k * ln(a) = ln(a^k)). These properties allow us to combine or simplify logarithmic expressions, which is crucial for isolating variables in equations.

Natural Logarithm (ln)

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately 2.71828. It is commonly used in calculus and algebra due to its unique properties, such as the fact that the derivative of ln(x) is 1/x. In solving equations involving ln, it is important to recognize how to manipulate and interpret these logarithmic expressions to find exact solutions.

Exponential Equations

Exponential equations involve variables in the exponent and can often be solved by rewriting them in logarithmic form. For instance, if ln(a) = b, then a = e^b. This relationship is crucial when dealing with equations that include logarithms, as it allows us to convert the logarithmic expressions back into exponential form to isolate and solve for the variable.