Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. e^x=5.7

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in the form f(x) = a * b^x, where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent. These functions exhibit rapid growth or decay and are characterized by their unique property that the rate of change is proportional to the function's value. Understanding exponential functions is crucial for solving equations like e^x = 5.7.

Natural Logarithms

Natural logarithms, denoted as ln(x), are the inverses of exponential functions with base 'e' (approximately 2.718). They are used to solve equations involving exponential growth or decay by allowing us to isolate the exponent. For example, to solve e^x = 5.7, we can take the natural logarithm of both sides, leading to x = ln(5.7).

Calculator Usage for Logarithms

Using a calculator to evaluate logarithmic expressions is essential for obtaining numerical solutions to equations. Most scientific calculators have dedicated buttons for natural logarithms (ln) and common logarithms (log). After isolating the variable using logarithms, you can input the expression into the calculator to find a decimal approximation, which is often required for practical applications.