Problem 61 - College Algebra Essentials

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. $$5 \ln (2 x)=20$$

Step-by-Step Solution

Answer

The exact solution for \(x\) is \( \frac{e^4}{2}\), which is approximately 27.47 when converted to decimal form

Step 1: Reorganize the equation and isolate the logarithmic expression

First, we must isolate \(\ln(2x)\) by dividing both sides of the equation by 5. We will then get: \n\(\ln(2x) = 4\)

Step 2: Convert the log equation to exponential form

Next, we rewrite the equation in exponential form. With natural logarithms (ln), the base is \(e\). So the equation \(\ln(2x) = 4\) is equivalent to:\n\(e^4 = 2x\)

Step 3: Solve for \(x\)

To solve for \(x\), we divide both sides of the equation by 2. Therefore, \(x = \frac{e^4}{2}\)

Step 4: Check for domain validity

The domain of the original logarithmic expression is \(x > 0\). Since \(e^4 > 0\), and the two is also positive, the result is a positive value, which falls into the domain, making the solution valid.

Step 5: Compute decimal approximation

Using a calculator, we find the decimal approximation of \(x\), correct to two decimal places. Hence, \(x \approx 27.47\)