NATURAL LOGSUnit 6

For x 0 and 0 a 1, y = loga x if and only if x = a y.The function given by f (x) = loga x is called the logarithmic function with base a.

Every logarithmic equation has an equivalent exponential form: y = loga x is equivalent to x = a y

A logarithmic function is the inverse function of an exponential function.

Exponential function: y = ax

Logarithmic function: y = logax is equivalent to x = ay

A logarithm is an exponent!

The function defined by f(x) = loge x = ln x

is called the natural logarithm function.

y = ln x

(x 0, e 2.718281)

y

x5

–5

y = ln x is equivalent to e y = x

In Calculus, we work almost exclusively with natural logarithms!

01ln

1ln e

EXAMPLES

32ln yx 32 lnln yx yx ln3ln2

2

3

lnyx 22

3

lnln yx yx ln2ln23

EXAMPLES

5

432lnz

yx

xy ln4ln

yx ln3ln221

zyx ln5ln4ln32ln

4lnxy

32ln yx

Derivative of Logarithmic FunctionsThe derivative is

'( )(ln ( ) )(

.)

d f xf xdx f x

2Find the derivative of ( ) ln 1 .f x x x

2

22

2

( 1)(ln 1)

12 1

1

d x xd dxx xdx x x

xx x

Example:

Solution:

Notice that the derivative of expressions such as ln|f(x)| has no logarithm in the answer.

EXAMPLE

3ln xy xln3

xxy 313'

EXAMPLE

3ln 2 xy 32 xu

xdu 2

32' 2

x

xy

EXAMPLE

xxy ln

Product Rule

1ln1' xx

xy

xy ln1'

EXAMPLE

23

1ln xy x1ln23

xxy

223

11

23'

EXAMPLE

11ln

xxy 1ln1ln

21

xx

11

11

21'

xxy

12

21' 2x

y

11' 2

x

y

EXAMPLE

xy secln

xx

xxy tansec

tansec'

EXAMPLE

xxy tansecln

xxxxxy

tansecsectansec'

2

xxx

xxxy sectansec

sectansec'

INTEGRATING IS GOING BACKWARDS Finding the anti-derivative using

natural logs is fun, fun, fun

35x

dx 3xudxdu

udu5

Cu ln5

Cx 3ln5

12

2xxdx 12 xu

xdxdu 2

udu

Cu ln

Cx 1ln 2

xdx

1xu 1

dxx

du2

1

duu

u 12

uu ln2

Cxx 1ln212

dxdux 2

xu 1

dxduu )1(2

duu112

2

2sin23

cos4

dsin23uddu cos2ddu cos42

1,2

uat

5,2

uat 5

1

2udu

uln2

1ln5ln2

5ln2

xdxtan

dxxx

cossin

xu cos

xdxdu sin

udu

CxCu coslnln

Cxxdx coslntan

INTEGRALS OF 6 BASIC TRIG FUNCTIONS

Cuudu sinlncot

Cuuudu tanseclnsec

Cuuudu cotcsclncsc

Cuudu coslntan

Cuudu sincosCuudu cossin

6

0

2tan

xdxxu 2dxdu 2

3

0

tan21

udu

3

0cosln

21

u

0cosln

3cosln

21

1ln

21ln

21

2ln1ln21

2ln21