Final answer:
The domain of ln(cos(x)) is defined for intervals where cos(x) is positive, and the range is all real numbers.
Step-by-step explanation:
The question asks to define the domain and range of the function ln(cos(x)). For logarithmic functions, the input argument must be positive, hence for cos(x) to be positive, x has to be within the intervals where the cosine function is above zero. This means that the domain of ln(cos(x)) is restricted to the intervals between (2k\u03c0, (2k+1)\u03c0) for all integers k. The cosine function oscillates between -1 and 1, but since the natural logarithm is undefined for non-positive numbers, x cannot take values where cos(x) \u2264 0. Therefore, the range of ln(cos(x)) is all real numbers since the natural logarithm can take any positive real number as its argument resulting in all real numbers as the output.