Logarithm Rules cheat sheet (M J Rhoades)
Notes: When log b is written, it is for any log base even base "e"
When "ln" is written it means base "e" which is log base 2.718283
If just log is written, with no subscript, it implies base 10
When ln is written it means base "e"
Rules
Algebra
Log b (x) = N
means
bN = x
log b (0) is undefined
log b b = 1
log b 1 = 0
log b bx = x
or
ln (ex) = x
log log bx = x
log b (xy) = log b (x) + log b y and vice versa
log b (x r) = r log b x
log b
= log b (x) - log b (y) and vice versa
Inverse function of logs: example
(x) = bx
and
g (x) = log b x
Inverse exponent functions of ln
(
-1
-1
(x)) = e ln (x) = x
( (x)) = ln (ex) = x
Special logs x = log e x and log x = log 10 x natural and common logs
Change of base formula log b (x) =
Trig
log b tan (x) = log b sin (x) - log b cos (x)
log b cot (x) =
Calculus
=1
(x) = ln (x) and taking the derivative
(x) =
or
ln (x) =
dx = (x) (ln (x) - 1) + c
= ln (x) +c
= (x - 1) ex + c
= ln ln (x) + c
ln (x) dx = x n + 1
(ln (x)) m dx =
+cn
(ln x) m -
(ln x) m - 1 dx n
1
dx = ln x + c
=
dx
dx
when g (x) = Tan (x)
dx