Say we're interested in a quantity $Q$ which is the area under a function $f(x)$ in a linear plot of $f(x)$ vs. $x$. I heard my professor say that if we plot $xf(x)$ in a semilog x plot, the area under the curve in this plot preserves the quantity $Q$. I cannot see how this can be true. Does the fact the derivative of $\ln(x)$ is $1/x$ play a role?

By "semilog x", I believe she means a plot linear on the y-axis and logarithmic on the x-axis.