Logarithmic function and $x$

Hi I was wondering why in a logarithm $x$ cannot be a negative number, since for the inverse graph I drew the $x$ values are only positive. In the question it asks why the first four points of the exponential function are imaginary in the logarithm.

• The argument can be a negative number. The result will not be in $\mathbb R$ though. – rubik Jan 3 '16 at 21:25
• What's your definition of $\log$? – user137731 Jan 3 '16 at 21:25

The graphical methods are an excellent way to represent this. However, from an algebraic point of view, if you have y = $\log_2(x)$

this means that $2^y$ = x. Now think about it, 2 to the power of any number will never return a negative value.

From the graph in the other answer, you can see the following: As y -> -∞ then x -> Infinitesimal value.

As for the inverse function, as x -> -∞ then y -> Infinitesimal value.

If you define $\log_a(x)$ as the inverse function of $a^x$ where $a\ge 0$ then it should be clear that because the range of $a^x$ is all positive numbers that the domain of $\log_a(x)$ is all positive numbers.

Here's a plot where $a=2$.

The logarithm is the inverse function of the exponential function, $a^x, a\geq 0$ which takes on positive values. Reflect this in the line $y = x$ and see the result.