Solve each equation. Give solutions in exact form. See Examples 5–9. ln(7 - x) + ln(1 - x) = ln (25 - x)

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 for solving equations involving exponential growth or decay. Recognizing that ln(x) is only defined for x > 0 is important when solving equations, as it sets constraints on the values that x can take.

Exponential Equations

Exponential equations involve variables in the exponent and can often be solved by rewriting them in logarithmic form. For instance, if we have an equation of the form e^x = a, we can take the natural logarithm of both sides to solve for x. This concept is vital when dealing with equations that have been transformed through logarithmic properties, as it allows for the extraction of the variable.