FAIL Logarithms

Question

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

Answer 1

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.

Answer 2

One of the properties of logarithms is that $$\log_wx=\frac{\log_qx}{\log_qs}$$ therefore $$\log_1 x = \frac{\log x}{\log 1} = \frac{\log x}0$$ and division by 0 is undefined. Therefore $\log_1x$ is nonsense.