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. 5 ln(2x)=20

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are defined for positive real numbers. The natural logarithm, denoted as ln, specifically uses the base 'e' (approximately 2.718). Understanding how to manipulate logarithmic expressions is crucial for solving logarithmic equations, as it involves properties such as the product, quotient, and power rules.

Domain of Logarithmic Functions

The domain of a logarithmic function is restricted to positive values. For the expression ln(2x), the argument (2x) must be greater than zero, which implies that x must be greater than zero. Recognizing and applying these domain restrictions is essential when solving logarithmic equations to ensure that any solutions found are valid within the context of the original problem.

Solving Logarithmic Equations

To solve logarithmic equations, one typically isolates the logarithmic term and then exponentiates both sides to eliminate the logarithm. In the given equation, 5 ln(2x) = 20, dividing both sides by 5 simplifies the equation, allowing for further manipulation. After finding the exact solution, using a calculator to approximate the result to two decimal places is often required for practical applications.