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. 5e^x=23

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions in which a variable appears in the exponent. To solve these equations, one typically isolates the exponential term and then applies logarithmic functions to both sides. This process allows for the transformation of the equation into a linear form, making it easier to solve for the variable.

Natural Logarithms

Natural logarithms, denoted as ln(x), are logarithms with the base 'e', where 'e' is approximately equal to 2.71828. They are particularly useful in solving exponential equations because they can effectively reverse the effect of the exponential function. When applied to an equation, natural logarithms help isolate the variable in the exponent, facilitating the solution process.

Calculator Use for Approximations

Using a calculator to obtain decimal approximations is essential in solving exponential equations, especially when the solutions involve logarithms. Most scientific calculators can compute natural and common logarithms, allowing students to convert their exact solutions into decimal form. This step is crucial for providing a practical answer that can be easily interpreted and applied in real-world contexts.