\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
Consider the function \(f(x) = a \frac{1}{x}\).
Find the value of \(a\) such that \(f(x)\) is a probability density function on the interval \([1, 2]\).
\FieldText{1}{$$\begin{aligned}[t] 1 = \int_{1}^{2} f(x) \, dx\\ 1&= \int_{1}^{2} a \frac{1}{x} \, dx \\1&= a \int_{1}^{2} \frac{1}{x} \, dx \\1&= a \left[ \ln(x) \right]_{1}^{2} \\1&= a \left( \ln(2) - \ln(1) \right)\\1&= a \ln(2) \\a &= \frac{1}{\ln(2)}\end{aligned}$$}