Problem 65 - College Algebra

Question

Solve each logarithmic equation in Exercises \(49-92 .\) 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. $$ 6+2 \ln x=5 $$

Step-by-Step Solution

Answer

The solution to the equation \(6+2 \ln x=5\) is \(x = e^{-1/2}\) which is approximately 0.61 when rounded to two decimal places.

Step 1: Isolate the logarithm

First, we have to isolate the logarithmic term. So, start by subtracting 6 from both sides of the equation: \(2 \ln x = 5 - 6\) which simplifies to \(2 \ln x = -1\)

Step 2: Get rid of coefficient of logarithm

Next, divide both sides by 2 to get rid of the 2 that is multiplied with the logarithm. This results into: \(\ln x = -1/2\)

Step 3: Convert the logarithmic equation into an exponential equation

We then rewrite the equation in exponential form. Remember that \(\ln x\) is log base \(e\) of \(x\), so the equation becomes: \(e^{-1/2} = x\)

Step 4: Check and round off the solution

The solution must be within the domain of the original equation, i.e., \(x > 0\). If \(x = e^{-1/2}\) is positive, it's a valid solution. Calculate the decimal value of \(x\) and round it off to two decimal places if needed.