Questions: Solve the logarithmic equation. Be sure to reject any value of x that is not in the domain of the original logarithmic expression. 9 ln (8 x)=18 Solve the equation. What is the exact solution? Select the correct choice below and, if necessary, fill in the answer to complete your choice. A. The solution set is e^2/8. (Type an exact answer in simplified form. Use integers or fractions for any numbers in the expression.) B. There are infinitely many solutions. C. There is no solution. What is the decimal approximation to the solution? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is . (Type an integer or decimal rounded to two decimal places as needed.) B. There are infinitely many solutions. C. There is no solution.

$Solve the logarithmic equation. Be sure to reject any value of x that is not in the domain of the original logarithmic expression. 9 ln (8 x)=18 Solve the equation. What is the exact solution? Select the correct choice below and, if necessary, fill in the answer to complete your choice. A. The solution set is e^2/8. (Type an exact answer in simplified form. Use integers or fractions for any numbers in the expression.) B. There are infinitely many solutions. C. There is no solution. What is the decimal approximation to the solution? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is . (Type an integer or decimal rounded to two decimal places as needed.) B. There are infinitely many solutions. C. There is no solution.$

Transcript text: Solve the logarithmic equation. Be sure to reject any value of $x$ that is not in the domain of the original logarithmic expression. \[ 9 \ln (8 x)=18 \] Solve the equation. What is the exact solution? Select the correct choice below and, if necessary, fill in the answer to complete your choice. A. The solution set is $\left\{\frac{e^{2}}{8}\right\}$. (Type an exact answer in simplified form. Use integers or fractions for any numbers in the expression.) B. There are infinitely many solutions. C. There is no solution. What is the decimal approximation to the solution? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is $\square$ \}. (Type an integer or decimal rounded to two decimal places as needed.) B. There are infinitely many solutions. C. There is no solution.

Solution

Solution Steps

Step 1: Isolate the Logarithmic Term

Starting with the equation: \[ 9 \ln(8x) = 18 \] we divide both sides by 9 to isolate the logarithmic term: \[ \ln(8x) = \frac{18}{9} = 2 \]

Step 2: Exponentiate Both Sides

Next, we exponentiate both sides to eliminate the logarithm: \[ 8x = e^2 \]

Step 3: Solve for $ x $

Now, we solve for $ x $ by dividing both sides by 8: \[ x = \frac{e^2}{8} \]

Step 4: Check the Domain

The original logarithmic expression $ \ln(8x) $ requires that $ 8x > 0 $, which implies $ x > 0 $. Since $ \frac{e^2}{8} $ is positive, it is within the domain.

Step 5: Decimal Approximation

Calculating the decimal approximation of the exact solution: \[ x \approx 0.9236 \] Rounding to two decimal places gives: \[ x \approx 0.92 \]