Basic rules of logarithms

There is a lot of stuff on the web about logarithms. Here's a digest of the most essential facts. Seems no matter how much I use logs, my head still swims if I try to reason them out. Seems best to just memorize these few simple rules without thinking about them too much.

Nomenclature:
• log_b(x) → "log, base b, of x"
   log[b](x) is also recognizable when "b" is complicated
• ln(x) → natural log: "log, base e, of x"
• ld(x) or lb(x) → binary logarithm: "log, base 2, of x"
• Log(x) → "log, base 10, of x"
• log(x)  → common log: may be base e, base 10, or base 2
                   depending on location, application, context, etc.

Definitions:
• log_b(x) = y ⇔ x = b^y
• ƒ(x) = log_b(x)
   for {b∈ℝ | b > 0, b ≠ 1} and {x∈ℝ | x > 0}

Trivial identities:
• log_b(b) = 1
• log_b(1) = 0

Logarithm property of equality:
Ok, there really isn't such a property. Logs are exponents, so it's just the multiplication property of equality at work. None the less:
• a = b ⇔ log(a) = log(b)

Product Rule:
• log_b(xy) = log_b(x) + log_b(y)

Quotient Rule:
• log_b(x/y) = log_b(x) - log_b(y)

Power Rule:
• log_b(x^y) = y log_b(x)
• log_b(x^1/y) = log_b(x) / y

Change of Base Rule:
• log_b(x) = log_a(x) / log_a(b)

Derivatives:
• D{ln(x)} = 1 / x
• D{lg(x)} = 1 / (x ln(10))
• D{log_b(x)} = 1 / (x ln(b))

Complex:
• ln(-1) = πi
• lg(-1) = πi / ln(10)
• log_b(-1) = πi / ln(b)

A few handy corollaries:
• log_b(x) = 1 / log_x(b)
• log_b(x) / n = log_bⁿ(x)
• log_b(1/x) = log_1/b(x) = -log_b(x)
• b^log_a(x) = x^log_a(b)
• b^log_b(x) = x
• 10^lg(x) = x
• e^ln(x) = x
• log(a) × log(b) = log(b^log(a))
• log_b(a) × log_c(d) = log_c(a) × log_c(d) / log_c(b)
• log_b^p(x) = (1/p)log_b(x)
   for {b∈ℝ | b > 0, b ≠ 1} and {p∈ℝ | p ≠ 0}

Calculating the antilog:
Some calculators have an inverse– or anti–log key, many do not. After all, how far over back should calculator manufacturers bend for users who don't really understand logs?
• antilog_b(x) = b^x