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. 3e^5x=1977

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.

Logarithms

Logarithms are the inverse operations of exponentiation, allowing us to solve for the exponent in an exponential equation. There are two common types: natural logarithms (base e) and common logarithms (base 10). Understanding how to manipulate logarithmic properties, such as the product, quotient, and power rules, is essential for solving exponential equations effectively.

Calculator Use for Approximations

Using a calculator to obtain decimal approximations is crucial for providing practical solutions to exponential equations. After solving the equation symbolically, one can input the logarithmic results into a calculator to find numerical values. This step is important for applications where exact values are less useful than their decimal representations, especially in real-world contexts.