In Apostol's book "Introduction to Analytic Number Theory", Apostol says:
For any arithmetical function $f$ we define its derivative $f'$ to be the arithmetical function given by the equation
$$f'(n)=f(n)\log(n) \quad\text{for}\quad n\ge1.$$
My question is: He claims the derivative is an arithmetic function, but $\log(n)$ does not always give you a natural number output and an arithmetic function is defined on natural numbers. How can we know that if $f(n)$ is defined for $n$ then $f'(n)$ will also be if it truly is an arithmetic function for all $n$?
An arithmetic function maps the natural numbers to some subset of the complex numbers.
The range of $\log$ is all real numbers and the range of $f$ is, by definition, some subset of the complex numbers. So the function $f'(n)$ defined by a product of a real and complex number will also map to a subset of the complex numbers.