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Logarithms

Logarithms​

Definition:

log_a b = x means a^x = b
a > 0, a != 1, b > 0

Rules:

log_a(xy) = log_a x + log_a y
log_a(x/y) = log_a x - log_a y
log_a(x^k) = k log_a x
log_a b = 1 / log_b a
log_a b = log_c b / log_c a

Special values:

log_a 1 = 0
log_a a = 1
a^(log_a x) = x

Speed tricks:

  • Convert logs to exponentials if stuck.
  • Same base logs can be compared directly if base > 1.
  • If 0 < base < 1, comparison direction reverses.