Problem 63 - 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. $$6+2 \ln x=5$$

Step-by-Step Solution

Answer

The solution to \(6+2 \ln x = 5\) is approximately \(x ≈ 0.61\).

Step 1: Isolate the logarithmic expression

The first step is to isolate the logarithmic part of the equation on one side. To do so, subtract 6 from both sides of the equation: \(2 \ln x = 5 - 6\), which simplifies to \(2 \ln x = -1\).

Step 2: Solve for the variable contained within the logarithm

Next, divide the equation by 2 to get the \(ln x\) by itself: \(ln x = -1 / 2\).

Step 3: Convert from log to exponential form

To remove the logarithm, we can exponentiate both sides of the equation using the base \(e\) (since we have a natural logarithm). In other words, \(e^{ln x} = e^{-1/2}\). This results in \(x = e^{-1/2}\).

Step 4: Calculation and approximation of solution

As a final step, calculate \(e^{-1/2}\) to get the value of \(x\). If necessary, use a calculator to get a decimal approximation, rounded to two decimal places: \(x ≈ 0.61\). Note that in this example, the resulting \(x\) value is positive, which it must be in a logarithmic equation. If you were to get a negative result, you would reject that value, since you cannot take the logarithm of a negative number.