Problem 60 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$6 \ln (2 x)=30$$

Step-by-Step Solution

Answer

Based on the step-by-step solution, the exact answer of the logarithmic equation is \( x = \frac{e^{5}}{2} \), approximately \( x \approx 74.21 \) to two decimal places.

Step 1: Divide by 6

First of all, divide both sides of the equation by 6 to isolate the \( \ln \) on one side. This gives \( \ln (2x) = \frac{30}{6} = 5 \).

Step 2: Convert to exponential form

After separating the \( \ln \) on one side, convert the logarithmic equation into an equivalent exponential equation. To do that, use the property \( \ln a = b \) can be converted to \( e^b = a \) (since \( \ln \) is the logarithm to the base \( e \)). This gives \( e^{5} = 2x \).

Step 3: Solve for x

The last step is to solve for \( x \) by dividing both sides of the equation by 2. This gives \( x = \frac{e^{5}}{2} \).

Step 4: Calculate decimal approximation

Finally, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. This results to \( x \approx 74.21 \) when rounded to two decimal places, which is within the domain of the original logarithmic expressions.