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 to solve for $x$ when the variable is in the exponent, you can use the natural logarithm function, since the natural logarithm is the inverse of the exponential function with base $e$.
3. Apply the natural logarithm to both sides of the equation to isolate $x$: $\backslash ln\backslash left(e^\{x\}\backslash right)\; =\; \backslash ln(5.7)$.
4. Use the logarithmic identity $\backslash ln\backslash left(e^\{x\}\backslash right)\; =\; x$ to simplify the left side, giving $x\; =\; \backslash ln(5.7)$.
5. To find a decimal approximation, use a calculator to evaluate $\backslash ln(5.7)$ 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 these functions behave is essential for solving equations involving exponents.

Natural Logarithms

The natural logarithm, denoted ln(x), is the inverse function of the exponential function with base e. It allows us to solve equations where the variable is in the exponent by 'undoing' the exponential. For example, if e^x = a, then x = ln(a).

Using a Calculator for Approximations

After expressing the solution in logarithmic form, calculators are used to find decimal approximations. This involves inputting the logarithmic expression and rounding the result to the desired decimal places, ensuring practical and usable answers.