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

lny

x 22

3

lnln yx yx ln2ln2

3

EXAMPLES

5

432ln

z

yx

xy ln4ln

yx ln3ln22

1

zyx ln5ln4ln32ln

4ln

x

y

32ln yx

Derivative of Logarithmic Functions

The derivative is

'( )(ln ( ) )

(.

)

d f xf x

dx f x

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

2

22

2

( 1)(ln 1)

12 1

1

dx xd dxx x

dx x xx

x 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

3

2'

2

x

xy

EXAMPLE

xxy ln

Product Rule

1ln1

' xx

xy

xy ln1'

EXAMPLE

2

3

1ln xy x1ln2

3

xxy

22

3

1

1

2

3'

EXAMPLE

1

1ln

x

xy 1ln1ln

2

1 xx

1

1

1

1

2

1'

xxy

1

2

2

1'

2xy

1

1'

2

xy

EXAMPLE

xy secln

xx

xxy tan

sec

tansec'

EXAMPLE

xxy tansecln

xx

xxxy

tansec

sectansec'

2

x

xx

xxxy sec

tansec

sectansec'

INTEGRATING IS GOING BACKWARDS

Finding the anti-derivative using natural logs is fun, fun, fun

3

5

x

dx 3xu

dxdu

u

du5

Cu ln5

Cx 3ln5

1

22x

xdx 12 xu

xdxdu 2

u

du

Cu ln

Cx 1ln 2

x

dx

1

xu 1

dxx

du2

1

duu

u 12

uu ln2

Cxx 1ln212

dxdux 2

xu 1

dxduu )1(2

duu

112

2

2sin23

cos4

d

sin23u

ddu cos2ddu cos42

1,2

uat

5,2

uat

5

1

2u

du

uln2

1ln5ln2

5ln2

xdxtan

dxx

x

cos

sin

xu cos

xdxdu sin

u

du

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

tan2

1

udu

3

0cosln

2

1

u

0cosln

3cosln

2

1

1ln

2

1ln

2

1 2ln1ln

2

1 2ln

2

1