The Ungerecht Family Tree

Combining powers

nᵃnᵇ  = nᵃ⁺ᵇ
nᵃ/nᵇ = nᵃ⁻ᵇ
(nᵃ)ᵇ = nᵃᵇ
n⁻ᵃ   = 1/nᵃ

(Note: The square root sign, √, refers only to the positive root. Use ± to include both roots.)

Log is the inverse of a power

x = bᵉ,   e = logᵦ(x)  (b = base, 10 by default)
eg. 2³ = 8, log₂(8) = 3
log(1) = 0 (for any base), log(x) is undefined for x =< 0
logₐ(x) = logᵣ(x) / logᵣ(a)
        = log(x) / log(a)  for r=10
        = ln(x) / ln(a)    for r=e  (e = Euler's number, 2.718..)
        
natural log: ln(x) = logₑ(x)
ln(e) = 1, log(10) = 1
eᵏ = c, and k = ln(c) => e^ln(c) = c

a^logₐ(x) = x   (power and log are inverses, cancel each other out)
logₐ(aᵏ) = k   (same reason)
product rule: log(ab) = log(a) + log(b)
quotient rule: log(a/b) = log(a) - log(b)
power rule: log(aᵇ) = b*log(a)

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