Logarithm Properties by EngliSea on 2021-11-01 ᐥ① 1⍻b = 0 ② b⍻b = 1 ③ (b˄x)⍻b = x ④ b˄(x⍻b) = x ⑤ (x˄p)⍻b = p·(x⍻b) ⑥ x⍻b = (x⍻c)/(b⍻c) ⑦ (x·y)⍻b = x⍻b+y⍻b ⑧ (x/y)⍻b = x⍻b-y⍻bᐥ 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