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

Verified step by step guidance

1. Start with the given exponential equation: \$5e^{x} = 23$.
2. Isolate the exponential expression by dividing both sides of the equation by 5: \(e^{x} = \frac{23}{5}\).
3. To solve for \(x\), take the natural logarithm (ln) of both sides, since the base of the exponential is \(e\): \(\ln\left(e^{x}\right) = \ln\left(\frac{23}{5}\right)\).
4. Use the logarithmic identity \(\ln\left(e^{x}\right) = x\) to simplify the left side: \(x = \ln\left(\frac{23}{5}\right)\).
5. To find a decimal approximation, use a calculator to evaluate \(\ln\left(\frac{23}{5}\right)\) and round the result to two decimal places.

Key Concepts

Exponential Equations

An exponential equation is one in which the variable appears in the exponent, such as 5e^x = 23. Solving these equations often requires isolating the exponential expression before applying logarithms to solve for the variable.

Natural Logarithms

The natural logarithm, denoted ln, is the inverse function of the exponential function with base e. It allows us to solve equations involving e^x by rewriting e^x = a as x = ln(a), facilitating the isolation of the variable in the exponent.

Using Calculators for Decimal Approximations

After expressing the solution in logarithmic form, calculators are used to find decimal approximations. This step involves evaluating logarithms and rounding the result to a specified number of decimal places, such as two decimals, for practical use.