What is the domain of f(x) = ln(x - 4)?

Question

Accepted Answer

x > 4. The natural logarithm function ln(u) is defined only when u > 0 (strictly greater than zero, not equal to zero). For f(x) = ln(x - 4), we need x - 4 > 0, which gives x > 4. The domain is (4, infinity).

"x >= 4" (x >= 4) is incorrect because at x = 4, f(4) = ln(0), which is undefined (ln(0) = -infinity, which is not a real number). "x > 0" (x > 0) would be the domain of ln(x) without the shift. "All real numbers" is wrong because the logarithm requires a positive argument. The key distinction is that logarithms require strictly positive arguments  -  zero is not allowed.

Domain restrictions for logarithmic functions are commonly tested on the CLEP exam. For ln(expression) or log(expression), set the argument strictly greater than zero: expression > 0. This is different from square root functions, which allow the radicand to be zero (sqrt(0) = 0 is fine). Also remember that the base of a logarithm must be positive and not equal to 1. When finding domains of composite functions, apply all restrictions from each component function.

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