Question

What is the range of f(x)=ln(cosx)? Why? Please do not use a calculator.

Answer 1

Well, first of all remember that ln(t) only makes sense for t > 0 and hence in this case for cos x > 0

Now cos x in itself has range [-1,1]. But in this function, the only part of that range we want is (0,1]. And for that interval now as the domain of ln, ln has a range of -infinity to 0,
\[(-\infty, 0]\]

Answer 2

Thank you!

Answer 3

A graph will help you. Draw one first of cos x over a range of -2pi to 2pi. Then ln(cos x) where it exists. This is a slightly exotic function but it will turn out, quite important for a particular application.