Problem 59 - 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. $$5 \ln (2 x)=20$$

Step-by-Step Solution

Answer

The exact answer to the equation is \(x = e^4/2\), and by using a calculator, a decimal approximation is \(x = 27.18\)

Step 1: Isolate the Logarithm

Firstly, isolate the logarithm on one side of the equation to simplify further calculations. Now, divide both sides of the equation by 5, obtaining the equation as: \(\ln(2x) = 4\)

Step 2: Convert to Exponential Form

AudioNow, convert this logarithmic equation into its equivalent exponential form. As a reminder, in general, if \(\ln(u) = v\), then \(e^v = u\), where \(e\) is Euler's number. Therefore, the equation \(\ln(2x) = 4\) converts to \(e^4 = 2x\).

Step 3: Solve for x

Now, solve this equation for \(x\). You can do this by dividing both sides of equation by 2. So, the solution is \(x = e^4/2\)