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

Verified step by step guidance

1. Identify the given exponential equation: \(e^{x} = 5.7\).
2. Recall that the natural logarithm function \(\ln(x)\) is the inverse of the exponential function with base \(e\). This means applying \(\ln\) to both sides will help isolate \(x\).
3. Apply the natural logarithm to both sides of the equation: \(\ln(e^{x}) = \ln(5.7)\).
4. Use the logarithmic identity \(\ln(e^{x}) = x\) to simplify the left side, resulting in \(x = \ln(5.7)\).
5. To find a decimal approximation, use a calculator to evaluate \(\ln(5.7)\) and round the result to two decimal places.

Key Concepts

Exponential Functions

An exponential function has the form f(x) = a^x, where the variable is in the exponent. In this problem, the base is the constant e (Euler's number, approximately 2.718), making it a natural exponential function. Understanding how to work with exponential functions is essential to isolate the variable in the exponent.

Natural Logarithms

The natural logarithm, denoted ln(x), is the inverse function of the exponential function with base e. Applying the natural logarithm to both sides of an equation like e^x = 5.7 allows you to solve for x by 'undoing' the exponentiation, since ln(e^x) = x.

Using a Calculator for Approximations

After expressing the solution in terms of logarithms, a calculator is used to find a decimal approximation. This involves evaluating the logarithm (e.g., ln(5.7)) and rounding the result to the desired precision, here to two decimal places, to provide a practical numerical answer.