Question

solve the following exponential equation. express the solution set in terms of natural logarithms or common logarithms. then, use a calculator to obtain a decimal approximation for the solution.\\( e^{1 - 6x} = 247 \\)\\( \\)\\( \\)the solution set expressed in terms of logarithms is \\( \boxed{} \\).\\( \\)\\( \\) (use a comma to separate answers as needed. simplify your answer. use integers or fractions for any numbers in the expression. use \\( \ln \\) for natural logarithm and \\( \log \\) for common logarithm.)

Explanation:

Step1: Take natural log of both sides

Take the natural logarithm of both sides of the equation \( e^{1 - 6x}=247 \). By the property of logarithms \( \ln(e^y)=y \), we get:
\( \ln(e^{1 - 6x})=\ln(247) \)
Simplifying the left - hand side, we have:
\( 1-6x = \ln(247) \)

Step2: Solve for x

First, subtract 1 from both sides of the equation:
\( - 6x=\ln(247)-1 \)
Then, divide both sides by - 6:
\( x=\frac{1 - \ln(247)}{6} \)

Now, we calculate the numerical value. We know that \( \ln(247)\approx5.517 \)
\( x=\frac{1 - 5.517}{6}=\frac{-4.517}{6}\approx - 0.753 \)

Answer:

In terms of natural logarithms, the solution is \( x = \frac{1-\ln(247)}{6} \), and the decimal approximation is \( x\approx - 0.753 \)