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^2x−3e^x+2=0

Exponential equations are mathematical expressions in which variables appear as exponents. To solve these equations, one often needs to isolate the exponential term and apply logarithmic functions. Understanding the properties of exponents, such as the fact that if a^m = a^n, then m = n, is crucial for finding solutions.

Natural logarithms (ln) and common logarithms (log) are two types of logarithmic functions used to solve exponential equations. The natural logarithm is based on the constant e (approximately 2.718), while the common logarithm is based on 10. These logarithms help to transform exponential equations into linear forms, making them easier to solve.

Using a calculator to obtain decimal approximations is essential in solving exponential equations, especially when exact solutions are complex or not easily expressible. Calculators can compute values of natural and common logarithms, allowing for quick conversions of logarithmic results into decimal form, which is often required for final answers.