Yes, the exponential function:

$f(x) = \exp(x) = e^x$ is defined for ALL real numbers, but only takes POSITIVE values.

Therefore, the natural logarithm:

$g(x) = \ln(x)$

is only defined for positive values, but has as range the entire real numbers (if we want a really "large negative" number, we take the log of something close to 0, if we need a really large number, we take the log of (a generally much) larger positive number).

Similar considerations hold for other "bases", since these can be expressed as suitable modifications of the "natural" bases:

$a^x = (e^{\ln(a)})^x = e^{x\ln(a)}$

$\log_a(x) = \dfrac{\ln(x)}{\ln(a)}$