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^(1−5x)=793

Verified step by step guidance

1. Start with the given exponential equation: \(e^{(1 - 5x)} = 793\).
2. To solve for \(x\), take the natural logarithm (ln) of both sides to utilize the property that \(\ln(e^y) = y\). This gives: \(\ln\left(e^{(1 - 5x)}\right) = \ln(793)\).
3. Simplify the left side using the logarithm property: \$1 - 5x = \ln(793)$.
4. Isolate the term containing \(x\) by subtracting 1 from both sides: \(-5x = \ln(793) - 1\).
5. Finally, solve for \(x\) by dividing both sides by \(-5\): \(x = \frac{1 - \ln(793)}{5}\).

Key Concepts

Exponential Equations

An exponential equation involves variables in the exponent, such as e^(1−5x) = 793. Solving these requires isolating the exponential expression and then applying logarithms to both sides to solve for the variable.

Natural Logarithms

The natural logarithm (ln) is the inverse function of the exponential function with base e. Applying ln to both sides of an equation like e^(1−5x) = 793 allows you to simplify the exponent and solve for x.

Using a Calculator for Approximations

After expressing the solution in logarithmic form, a calculator is used to find decimal approximations. This step involves evaluating logarithms and performing arithmetic to get a numerical answer rounded to the desired decimal places.