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

Verified step by step guidance

1. Recognize that the equation is given as \(\ln x = 2\), where \(\ln\) denotes the natural logarithm, which is the logarithm with base \(e\) (Euler's number).
2. Rewrite the logarithmic equation in its equivalent exponential form. Recall that if \(\ln x = 2\), then \(x = e^2\).
3. Express the solution as \(x = e^2\), which is the exact form of the answer.
4. Check the domain of the original logarithmic function. Since \(\ln x\) is defined only for \(x > 0\), verify that \(e^2\) is positive, which it is, so no values are rejected.
5. If a decimal approximation is needed, use a calculator to evaluate \(e^2\) and round the result to two decimal places.

Key Concepts

Properties of Logarithms

Understanding the natural logarithm function ln(x) is essential, where ln(x) is the logarithm base e. Key properties include ln(e) = 1 and the ability to rewrite logarithmic equations in exponential form, such as ln(x) = 2 becoming x = e^2.

Domain of Logarithmic Functions

The domain of ln(x) is x > 0, meaning the argument inside the logarithm must be positive. When solving equations, any solution that results in a non-positive argument must be rejected to ensure the solution is valid within the function's domain.

Converting Between Logarithmic and Exponential Forms

Solving logarithmic equations often requires rewriting them in exponential form. For example, ln(x) = 2 can be rewritten as x = e^2, which allows direct computation of x. This conversion simplifies solving and interpreting logarithmic equations.