In every lower level math textbook that I have seen (up through at least Calculus), log refers to the “common logarithm” (base 10—which was admittedly much more common before electronic calculators were readily available). But in more advanced math (e.g. discussion of the Prime Number Theorem) log is often used instead of ln to denote the natural logarithm (base e).
Every scientific calculator I’ve ever seen has both a log button (for log[sub]10[/sub]) and an ln button. But (as ZenBeam noted), log gives the natural logarithm in many computer programming contexts.
To make matters even more interesting, when dealing with complex variables, Log or Ln is capitalized to denote the principal branch of the logarithm function.
I’ve always pronounced “ln(x)” as “el en of x” or “the natural log of x.”