Remember that $\log_a b = x$ is really just $a^x = b$ in disguise. Therefore, you have

$$6^x = 1$$

So basically, what values of $x$ will make the answer $1$? Well, it's gotta be $0$.

So from this you can deduce that any which looks like

$$\log_x 1$$

must have the answer zero or in other words,

$$\log_x 1 = 0$$

Because $6^0=1$. Simple as that. In fact, $\log_a 1=0$ for any value of $a>0$.

The $\log$ function is the inverse function to the exponential function. Thus, the number $x=\log_a b$ is the number that solves the equation $a^x = b$.

Apply this to your example: what is $x=\log_6 1$? To what power must you put $6$ to get $1$? Well, you know that $6^0 = 1$, right? this means that by definition, $0 = \log_6 1$.

$$\log_6\frac11=\log_61-\log_61.$$

$\log_b (xy) = \log_b x + \log_b y$. Since $1x = x$, we must have $\log_b 1 = 0$.