Let's solve the equation [tex]\(3 \ln(x) = 12\)[/tex] step-by-step.
Step 1: Start with the given equation:
[tex]\[ 3 \ln(x) = 12 \][/tex]
Step 2: Divide both sides of the equation by 3 to isolate [tex]\(\ln(x)\)[/tex].
[tex]\[ \frac{3 \ln(x)}{3} = \frac{12}{3} \][/tex]
[tex]\[ \ln(x) = 4 \][/tex]
Step 3: Rewrite the equation in its exponential form to solve for [tex]\(x\)[/tex]. The natural logarithm [tex]\(\ln(x)\)[/tex] is the power to which [tex]\(e\)[/tex] (the base of natural logarithms) must be raised to get [tex]\(x\)[/tex]. Therefore, [tex]\(\ln(x) = 4\)[/tex] can be written as:
[tex]\[ x = e^4 \][/tex]
Conclusion: The value of [tex]\(x\)[/tex] is [tex]\(e^4\)[/tex]. Given the precise numerical value from computations, [tex]\(e^4 \approx 54.598150033144236\)[/tex].
Thus, the complete solution is:
[tex]\[ \ln(x) = 4 \][/tex]
[tex]\[ x \approx 54.598150033144236 \][/tex]
So, the solution to the equation [tex]\(3 \ln(x) = 12\)[/tex] is:
[tex]\[ x \approx 54.598150033144236 \][/tex]
