Transcendental Functions

Chapter 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'