Question

what is the approximate solution to this equation?
$4\ln x - 8 = 12$

a. $x \approx 10.5$
b. $x \approx 148.4$
c. $x \approx 1.61$
d. $x \approx 59,874$

Explanation:

Step1: Isolate the logarithmic term

Start with the equation \(4\ln x - 8 = 12\). Add 8 to both sides to isolate the term with the logarithm:
\(4\ln x = 12 + 8\)
\(4\ln x = 20\)

Step2: Solve for \(\ln x\)

Divide both sides by 4 to solve for \(\ln x\):
\(\ln x = \frac{20}{4}\)
\(\ln x = 5\)

Step3: Exponentiate to solve for \(x\)

Recall that \(\ln x = y\) is equivalent to \(x = e^y\). So, exponentiate both sides with base \(e\):
\(x = e^5\)

Step4: Approximate \(e^5\)

Calculate \(e^5\) (where \(e \approx 2.71828\)):
\(e^5 \approx 2.71828^5 \approx 148.413\)

Answer:

B. \( x \approx 148.4 \)