Does integrability of $f>0$ imply integrability of $\ln (f)$?

In problem 36, chapter 18 of Spivak's Calculus, he asks the following:

Prove that for all integrable $f>0$, we have $$\frac{1}{b-a}\int_a^b \ln f \le \ln \left(\frac{1}{b-a} \int_a^b f \right)$$

I am able to prove this provided I assume $\ln f$ is also integrable on $[a,b].$ Is this a mistake by the author or does the integrability of $f>0$ actually imply integrability of $\ln f$?