➊ 1⍻b = 0
The power that the base must be raised to in order to yield 1 is the 0th.
➋ b⍻b = 1
The power that the base must be raised to in order to yield the base itself is the 1st.
➌ (b˄x)⍻b = x
The power that the base must be raised to in order to yield the base raised to the power of a number is the power of the number.
➍ b˄(x⍻b) = x
A number raised to the power that the number must be raised to in order to yield x is x.
➎ (x˄p)⍻b
┆x=b˄(x⍻b)┆
= ((b˄(x⍻b))˄p)⍻b
┆(b˄♢)˄p=b˄(♢·p)┆
= (b˄((x⍻b)·p))⍻b
┆(b˄♡)⍻b=♡┆
= (x⍻b)·p
= p·(x⍻b)
x⍻c
┆x=b˄(x⍻b)┆
= (b˄(x⍻b))⍻c
┆(b˄♡)⍻c=♡·(b⍻c)┆
= (x⍻b)·(b⍻c)
➏ x⍻b = (x⍻c)/(b⍻c)
➐ (x·y)⍻b
┆x=b˄(x⍻b), y=b˄(y⍻b)┆
= ((b˄(x⍻b))·(b˄(y⍻b)))⍻b
┆(b˄♢)·(b˄♡)=b˄(♢+♡)┆
= (b˄(x⍻b+y⍻b))⍻b
┆(b˄♢)⍻b=♢┆
= x⍻b+y⍻b
➑ (x/y)⍻b
┆x=b˄(x⍻b), y=b˄(y⍻b)┆
= ((b˄(x⍻b))/(b∧(y⍻b)))⍻b
┆(b˄♢)/(b˄♡)=b˄(♢-♡)┆
= (b˄(x⍻b-y⍻b))⍻b
┆(b˄♡)⍻b=♡┆
= x⍻b-y⍻b
