Question

What is the domain of F(x) = In(x)?

Answer 1

Answer as an inequality:  $x > 0$

Answer in interval notation:  $(0, \infty)$

Answer in words: Set of positive real numbers

All three represent the same idea, but in different forms.

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Explanation:

Any log is the inverse of an exponential equation. Consider a general base b such that f(x) = b^x. The inverse of this is $f^{-1}(x) = \log_b(x)$

For the exponential b^x, we cannot have b^x = 0. We can get closer to it, but we can't actually get there. The horizontal asymptote is y = 0.

Because of this, $\log_b(x)$ has a vertical asymptote x = 0 (recall that x and y swap, so the asymptotes swap as well). This means we can get closer and closer to x = 0 from the positive side, but never reach x = 0 itself.

The domain of $\log_b(x)$ is x > 0 which in interval notation would be $(0, \infty)$. This is the interval from 0 to infinity, excluding both endpoints.

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The natural log function Ln(x) is a special type of log function where the base is b = e = 2.718 approximately.

So,

$\log_e(x) = \text{Ln}(x)$

allowing all of what was discussed in the previous section to apply to this Ln(x) function as well.

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In short, the domain is the set of positive real numbers. We can't have x be 0 or negative.