Cheat Sheet – Logarithm
Definition
Graph
x
a = N ⇔ x = log a N
f(x) = log a x ,
a>1
f(x) = log a x ,
0<a<1
(a > 0, a ≠ 1)
Properties
aloga N = N
log a a = 1
log a 1 = 0
log a M + log a N = log a MN
log a M − log a N = log a
M
N
log a M p = p log a M
1
log ap M = log a M
p
a > 0, a ≠ 1; M, N > 0
log a b =
log c b
log c a
a, b, c > 0; a, c ≠ 1
log a b =
1
log b a
a, b > 0; a, b ≠ 1
alogb c = c logb a
a, b, c > 0; b ≠ 1
Inequalities
If a > 1, then
0 < x < y ⇔ log a x < log a y
If 0 < a < 1, then
0 < x < y ⇔ log a x > log a y
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