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Logarithm 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 Limits

=1 Derivatives:

ƒ/ (ax) = ax ln (a) ƒ/ (ex) = ex ƒ/ (ln(x)) = , x > 0 ƒ/ (loga (x)) = ƒ/ (loga u) =

,x>0 (loga e)

Integrals:

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

dx = ln x + c =

dx

dx

when g (x) = Tan (x)

dx

1

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 Limits

=1 Derivatives:

ƒ/ (ax) = ax ln (a) ƒ/ (ex) = ex ƒ/ (ln(x)) = , x > 0 ƒ/ (loga (x)) = ƒ/ (loga u) =

,x>0 (loga e)

Integrals:

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

dx = ln x + c =

dx

dx

when g (x) = Tan (x)

dx

1