Solve each equation. Give solutions in exact form. See Examples 5–9. ln x + ln x^2 = 3

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is essential for solving equations involving logarithmic expressions. Key properties include the product rule, which states that ln(a) + ln(b) = ln(ab), and the power rule, which states that ln(a^b) = b * ln(a). These properties allow us to combine and simplify logarithmic terms effectively.

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. Understanding how to manipulate and solve equations involving ln is crucial for finding exact solutions.

Exponential Equations

Exponential equations are equations in which variables appear as exponents. To solve these equations, one often needs to rewrite them in logarithmic form. For example, if ln(x) = a, then x = e^a. This concept is vital for transforming logarithmic equations into a solvable format, especially when dealing with equations that involve natural logarithms.