Solve each logarithmic equation in Exercises 49–92. 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=2

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. The natural logarithm, denoted as ln, specifically refers to logarithms with base e (approximately 2.718). Understanding how to manipulate and solve equations involving logarithms is crucial, as they often appear in various mathematical contexts, including growth and decay problems.

Domain of Logarithmic Functions

The domain of a logarithmic function is the set of all positive real numbers. This means that for any logarithmic expression, the argument (the value inside the logarithm) must be greater than zero. When solving logarithmic equations, it is essential to check that any potential solutions fall within this domain to avoid invalid results.

Solving Logarithmic Equations

To solve logarithmic equations, one typically uses properties of logarithms to isolate the variable. For example, the equation ln x = 2 can be rewritten in exponential form as x = e^2. After finding potential solutions, it is important to verify that they are valid by ensuring they meet the domain restrictions of the original logarithmic expressions.