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I need help finding the domain and range of logarithm functions. For example, what is the domain and range of $y=\log(x-3)$?

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If $y=\log_ax$ where real $a>0;$ we have $x=a^y>0\implies x$ must be $x>0$ – lab bhattacharjee Aug 11 '13 at 18:03
Ok, why does this have $14197$ views? – bwv869 May 18 '14 at 15:26
@Oliver From google and other search engines. Someone gets a homework problem and wants to figure out how to "find the domain and range of a logarithmic function" and boom, this pops up. – Michael Tong May 18 '14 at 15:42
up vote 9 down vote accepted

Let's take a look at the function $$y = \log(x - 3)$$

We are trying to find the Domain and the Range of this function, recalling that:

  • Domain: Includes all values of $x$ for which the function is defined.
  • Range: Includes all values $y$ for which there is some $x$ such that $y = \log(x - 3)$.

    It makes no sense to write $y = \log(a)$ when $a \leq 0$ because $\log(a)$ is defined only for positive $a$. So in this problem, $y = \log(x = 3)$, is defined if and only if $x - 3 \gt 0 \iff x \gt 3$, and that gives you the domain $x\in (3, +\infty)$.

The range of $y$ is all of $\mathbb R$.

Graphing a function is a great way to confirm or gain insight into the domain and range of a function.

E.g. Below is the graph of the function $y = \log_e(x - 3)$.

enter image description here

As $x$ grows in size, so to does $y$. That is, $y$ is an increasing function. As $x$ approaches $3$ from the right, although $y=\log_e(x - 3)$ is not defined for any $x \leq 3$, within the interval $x \in (3, 4)$, $y < 0$. The closer $x$ is to $3$ (from the right), the smaller $y$ gets (the "more negative" $y$ becomes.)

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Remember that for $x \in \mathbb{R}$ the argument of $\log$ cannot be negative and $\log(x)$ varies from $-\infty$ to $\infty$

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$\log$ is the inverse function of exponentiation: $y = \log_b x \iff x = b^y$ (only for positive $b$). So the domain of $y = \log_b x$ is the range of $x = b^y$ which is all $x$ with $x > 0$ since $b^y$ is always positive for positive $b$. And the range of $y = \log_b x$ is the domain of $x = b^y$ which is any number $y \in \mathbb{R}$.

Now for other variations, just apply the above. In your example we have $y = \log(x - 3)$. the domain of $\log$ is positive numbers and so we must have $x - 3 > 0 \implies x > 3$. The range is still all real numbers.

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$y = \log(x-3)$



so domain is $(3,\infty)$

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