Problem 62 - 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. $$6 \ln (2 x)=30$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = e^5 / 2\) or \(x \approx 74.21\)

Step 1: Simplify the equation

Firstly, it is required to get the logarithmic expression on one side of the equation. The problem already provides this kind of set up: \(6 \ln (2x) = 30\). However, both sides of the equation can be divided by 6 to simplify it further: \(\ln (2x) = 5\).

Step 2: Convert the logarithm into exponential

As the base of the natural logarithm is \(e\), \(\ln (2x)\) can be written as an exponential equation \(e^5 = 2x\).

Step 3: Solve for \(x\)

Now, you can solve for \(x\). Divide both sides of the equation by 2 to isolate \(x\), so \(x = e^5 / 2\). Using a calculator, that can be approximated to \(x \approx 74.21\).

Step 4: Check that solution is valid

Since the logarithm function is undefined for negative values and zero, it is required to check if \(x\) = \(e^5 / 2\) fits within the domain. Since \(e^5 / 2\) is positive, it falls within the valid domain of the logarithmic function, hence \(x\) = \(e^5 / 2\) or \(x \approx 74.21\) is the solution of the original problem.