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, approximately 2.718).
2. Rewrite the logarithmic equation in its equivalent exponential form. Recall that \( \ln x = 2 \) means \( x = e^2 \).
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 \( e^2 > 0 \), which it is, so the solution is valid.
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 properties of logarithms, such as the definition of the natural logarithm (ln), is essential. The natural logarithm ln x is the inverse of the exponential function e^x, meaning ln x = y implies x = e^y. This relationship allows solving equations involving ln by rewriting them in exponential form.

Domain of Logarithmic Functions

The domain of a logarithmic function like ln x is all positive real numbers (x > 0). When solving logarithmic equations, it is crucial to check that the solutions fall within this domain, rejecting any values that make the argument of the logarithm zero or negative, as these are undefined.

Exact and Approximate Solutions

Logarithmic equations often yield exact solutions expressed in terms of constants like e. After finding the exact solution, it is common to use a calculator to approximate the value to a desired decimal place, ensuring clarity and practical usability of the answer.