Q:

FAIL Logarithms

$$\log_11^x=x$$
$$\log_11^y=y$$
$$1^x=1^y\implies\log_11^x=\log_11^y$$
$$\therefore x=y$$
But
$$1^7=1^8\implies7=8$$
Where is my error? And why?

A:

The logarithm function $\log_a$ is typically defined to be the inverse function of the exponential $a^x$. This makes sense for $a > 0$ and $a \ne 1$ since the exponential function is strictly increasing.
But $1^x$ is constant, and so doesn't have an inverse. Therefore, $\log_1$ doesn't really make sense, and it certainly can't be expected to have the same rule as usual logarithms.

