CHAPTER 5SECTION 5.5

BASES OTHER THAN e AND APPLICATIONS

Definition of Exponential Function to Base a

• a0 = 1

• axay = ax+y

• ax = ax-y

ay

• (ax)y = axy

Definition of Logarithmic Function to Base a

log log loga a axy x y

log log loga a a

xx y

y

log 1 0a

loglog xya yxxy aaa logloglog log logna ax n x

Properties of Inverse Functions

2

7

Solve for in each of the following equations.

1. log 4

12. 3

81

3. log 1

14. log

x

a

x

x

x

xa

2

7

Solve for in each of the following equations.

1. log 4

12. 3

81

3. log 1

14. log

x

a

x

x

x

xa

2

7

Solve for in each of the following equations.

1. log 4

12. 3

81

3. log 1

14. log

x

a

x

x

x

xa

2

7

Solve for in each of the following equations.

1. log 4

12. 3

81

3. log 1

14. log

x

a

x

x

x

xa

2

7

Solve for in each of the following equations.

1. log 4

12. 3

81

3. log 1

14. log

x

a

x

x

x

xa

Theorem 5.13 Derivatives for Bases Other Than e

Examples for Derivatives for Bases other than e

10ln

tan

cos10ln

sinsin

cosln10

1y'

sin ,cos ,10coslog c.

82ln3322ln'

3 ,3 ,22 b.

22ln'22 a.

3

3

x

x

xx

x

xdxduxuaxy

y

dxduxuay

yay

xx

x

xx

Examples for Derivatives for Bases other than e

10ln

tan

cos10ln

sinsin

cosln10

1y'

sin ,cos ,10coslog c.

82ln3322ln'

3 ,3 ,22 b.

22ln'22 a.

3

3

x

x

xx

x

xdxduxuaxy

y

dxduxuay

yay

xx

x

xx

Examples for Derivatives for Bases other than e

10ln

tan

cos10ln

sinsin

cosln10

1y'

sin ,cos ,10coslog c.

82ln3322ln'

3 ,3 ,22 b.

22ln'22 a.

3

3

x

x

xx

x

xdxduxuaxy

y

dxduxuay

yay

xx

x

xx

Examples for Derivatives for Bases other than e

10ln

tan

cos10ln

sinsin

cosln10

1y'

sin ,cos ,10coslog c.

82ln3322ln'

3 ,3 ,22 b.

22ln'22 a.

3

3

x

x

xx

x

xdxduxuaxy

y

dxduxuay

yay

xx

x

xx

Examples for Derivatives for Bases other than e

10ln

tan

cos10ln

sinsin

cosln10

1y'

sin ,cos ,10coslog c.

82ln3322ln'

3 ,3 ,22 b.

22ln'22 a.

3

3

x

x

xx

x

xdxduxuaxy

y

dxduxuay

yay

xx

x

xx

Examples for Derivatives for Bases other than e

10ln

tan

cos10ln

sinsin

cosln10

1y'

sin ,cos ,10coslog c.

82ln3322ln'

3 ,3 ,22 b.

22ln'22 a.

3

3

x

x

xx

x

xdxduxuaxy

y

dxduxuay

yay

xx

x

xx

Examples for Derivatives for Bases other than e

10ln

tan

cos10ln

sinsin

cosln10

1y'

sin ,cos ,10coslog c.

82ln3322ln'

3 ,3 ,22 b.

22ln'22 a.

3

3

x

x

xx

x

xdxduxuaxy

y

dxduxuay

yay

xx

x

xx

Examples for Derivatives for Bases other than e

10ln

tan

cos10ln

sinsin

cosln10

1y'

sin ,cos ,10coslog c.

82ln3322ln'

3 ,3 ,22 b.

22ln'22 a.

3

3

x

x

xx

x

xdxduxuaxy

y

dxduxuay

yay

xx

x

xx

Examples for Derivatives for Bases other than e

10ln

tan

cos10ln

sinsin

cosln10

1y'

sin ,cos ,10coslog c.

82ln3322ln'

3 ,3 ,22 b.

22ln'22 a.

3

3

x

x

xx

x

xdxduxuaxy

y

dxduxuay

yay

xx

x

xx

Examples for Derivatives for Bases other than e

10ln

tan

cos10ln

sinsin

cosln10

1y'

sin ,cos ,10coslog c.

82ln3322ln'

3 ,3 ,22 b.

22ln'22 a.

3

3

x

x

xx

x

xdxduxuaxy

y

dxduxuay

yay

xx

x

xx

Examples for Derivatives for Bases other than e

10ln

tan

cos10ln

sinsin

cosln10

1y'

sin ,cos ,10coslog c.

82ln3322ln'

3 ,3 ,22 b.

22ln'22 a.

3

3

x

x

xx

x

xdxduxuaxy

y

dxduxuay

yay

xx

x

xx

Examples for Derivatives for Bases other than e

10ln

tan

cos10ln

sinsin

cosln10

1y'

sin ,cos ,10coslog c.

82ln3322ln'

3 ,3 ,22 b.

22ln'22 a.

3

3

x

x

xx

x

xdxduxuaxy

y

dxduxuay

yay

xx

x

xx

Theorem 5.14 The Power Rule for Real Exponents

x

e

x

e

xy

xdx

d

edx

d

edx

d

Differentiate.

0

xe

1eex

Hint: Take the ln of both sides.

xxy lnln xxy lnln

)1(ln1'

xx

xy

y

xyy ln1'

xxy x ln1'

Find each derivative with respect to the given variable.

23

1.t

f tt

2

22. log1

xy

x

Find each derivative with respect to the given variable.

23

1.t

f tt

2

22. log1

xy

x

2 2

2

ln3 3 2 3 1'

t ttf t

t

2

2

3 2 ln3 1t t

t

2 22log log 1x x

1 1 1 12 1 1

ln2 ln2 1

dy

dx x x

2 1 1

ln2 ln2 1x x

1 2 1

ln2 1x x

1 2

ln2 1

x

x x

Find each derivative with respect to the given variable.

13. 2

xy x

Find each derivative with respect to the given variable.

13. 2

xy x

1 2 1xdy

x xdx

1ln ln 2

xy x

ln 1 ln 2y x x

1 11 ln 2

2

dyx x

y dx x

1ln 2

2

dy xy x

dx x

1 12 ln 2

2

xdy xx x

dx x

•Differentiate .

–Using logarithmic differentiation, we have:

ln ln ln

' 1 1(ln )

2

1 ln 2 ln'

2 2

x

x

y x x x

yx x

y x x

x xy y x

x x x

LOGARITHMIC DIFFERENTIATION

xy x

To integrate exponential functions other than base e, either

• Convert to base e using the formula

OrIntegrate directly using the integration formula

xax ea ln

Caa

dxa xx

ln

1

•

EXP. FUNCTIONS WITH BASE a Example 14

55

00

5 0

22

ln 2

2 2

ln 2 ln 231

ln 2

xxdx

xa dx Theorem:

Proof:xa dx

2

1. 5 xx dx

xa dx Theorem:

1

lnxa C

a

Proof:xa dx ln xae dx

lnu x a

lndu adx

lnx ae dx1

lnue du

a

1

lnue C

a ln1

lnx ae C

a

ln1

ln

xae Ca

1

lnxa C

a

2

1. 5 xx dx2u x

2du x dx

15

2u du 1

52ln5

u C

215

2ln5x C

Theorem 5.15 A Limit Involving e

Applications of Exponential Functions

• Compound Interest Formulas

– Compounded n times per year:

– Compounded continuously:

Where t = number of years,

P = Initial deposit,

A = balance after t years,

r = interest rate expresses as a decimal,

n = number of compounding periods per years.

nt

n

rPA

1

rtPeA

A deposit of $2500 invested into an account paying an interest rate of 10%. Find its balance after 5 years if interest is compounded…a. quarterly b. monthly c. continuously

A deposit of $2500 invested into an account paying an interest rate of 10%. Find its balance after 5 years if interest is compounded…a. quarterly b. monthly c. continuously

$4113.27

.10 52500A e

$4121.80

4 5.10

2500 14

A

$4096.54

12 5.10

2500 112

A