log base 1 of 1

What is $\log(1)$ to the base of $1$? My teacher says it is $1$. I beg to differ, I think it can be all real numbers! i.e., $1^x = 1$, where $x\in \mathbb{R}$.

So I was wondering where I have gone wrong.

Solution 1:

The reason why it is not convenient to define $\log$ for the base of $1$ is simple:

$$\log_11=\frac{\log_e 1}{\log_e 1}$$

But the denominator is $0$ and thus the division doesn't make any sense unless we're working with limits :)

Solution 2:

What is $\dfrac00$? What number must $x$ be if $0\cdot x=0$? It can be any number.

What is $\log_1 1$? What number must $x$ be if $1^x=1$? It can be any number.

Hence these expressions are undefined.

What is $\lim\limits_{x\to a}\dfrac{f(x)}{g(x)}$ if $\lim\limits_{x\to a}f(x)=\lim\limits_{x\to a} g(x)=0$? In some cases it's $6$. It depends on which functions $f$ and $g$ are. It can be any number or $\infty$ or $-\infty$. But it's not always undefined. In many cases it's defined and equal to a particular number. For that reason $\dfrac00$ is an indeterminate form.

What is $\lim\limits_{x\to a}\log_{f(x)}g(x)$ if $\lim\limits_{x\to a}f(x)=\lim\limits_{x\to a} g(x)=1$? Again this depends on which functions $f$ and $g$ are. In many cases it's a specific number. This is also an indeterminate form.