Which of the following statements is FALSE? $\log _{5}(0.2)=-1$ $\log _{4}(4096)=6$ $\log _{2}(3)=8$ $\log (100)=2$ $\log _{6}(1)=0$

Step 1 ：We use the identity $a\log_b{x}=\log_b{x^a}$ and the definition of logarithm $\log_b{b}=1$ and $\log_b{1}=0$ to check each statement.

Step 2 ：First, we check $\log _{5}(0.2)=-1$. We know that $0.2=\frac{1}{5}$, so $\log _{5}(0.2)=\log _{5}(\frac{1}{5})=\log _{5}(5^{-1})=-1$, which is TRUE.

Step 3 ：Next, we check $\log _{4}(4096)=6$. We know that $4096=4^6$, so $\log _{4}(4096)=\log _{4}(4^6)=6$, which is TRUE.

Step 4 ：Then, we check $\log _{2}(3)=8$. We know that $3\neq2^8$, so $\log _{2}(3)\neq8$, which is FALSE.

Step 5 ：After that, we check $\log (100)=2$. We know that $100=10^2$, so $\log (100)=\log (10^2)=2$, which is TRUE.

Step 6 ：Finally, we check $\log _{6}(1)=0$. We know that $1=6^0$, so $\log _{6}(1)=\log _{6}(6^0)=0$, which is TRUE.

Step 7 ：So, the statement $\log _{2}(3)=8$ is FALSE.