Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30

Warm-UpWarm-Up

Solve for x in each equation.

a) 3x = b) log2x = –4

c) 5x = 30

Logarithmic, Exponential, and Other Transcendental Functions 20145

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Chapter 5Chapter 5

Transcendental Functions:Transcendental Functions:Bases Other than e

Day 1: All rules and derivative examples.Day 1: All rules and derivative examples.

Day 2: Integration examples and applications.Day 2: Integration examples and applications.

http://www.youtube.com/watch?v=SNZgbj3UaRE

Recall that the natural log, ln x, is a logarithm with base e. We can differentiate ln x simply with the following rule:

Bases Other than eBases Other than e

xx eedx

d

In addition, we know that the natural exponential function, ex, can easily be differentiated:

x

xdx

d 1ln

ueedx

d uu u

uu

dx

d ln

By applying the chain rule, we get:

and

But, what do we do if we have a logarithm or exponential function that has a base other than e? To determine these derivatives, we need to use a very useful logarithmic operation called the change of change of base formulabase formula. Recall from precalculus:

Bases Other than eBases Other than e

a

b

a

bb

e

ea ln

ln

log

loglog

This formula allows us to transform any logarithm into a quotient of logarithms with any base that we choose, including the natural logarithm. More specifically:

a

bb

c

ca log

loglog

a

x

dx

d

ln

ln

Now let’s try to find the derivative of a logarithm with a base other than e:

Bases Other than eBases Other than e

xdx

dalog

This is a constant

xa

1

ln

1 x

dx

d

aln

ln

1

Bases Other than eBases Other than e

To determine the derivative of a natural exponential function with a base other than e, we need to note:

axedx

d ln xadx

d

ae ax lnln

axax eeax lnln

xaaln

l nu a axu ln

ueedx

d uu

So, how do we integrate an exponential function with a base other than e? Once again, we use the alternate form of the exponential function of ax:

In terms of integration, at this point, we cannot find the integral of ln x, but we can integrate ex. Recall:

Bases Other than eBases Other than e

Cedxe xx

Cea

u ln

1 duea

u

ln

1

dxe ax ln dxa x

Cea

ax ln

ln

1

axax eeax lnln

1

lnxa C

a

lnd u a d xaxu ln

Caa

dua uu ln

1.6

The rules to differentiating with bases other than e

Rules for derivatives of bases Rules for derivatives of bases Other than eOther than e

xa

xdx

da

1

ln

1log.1

Caa

dxa xx ln

1.5

u

u

au

dx

da

ln

1log.2

xx aaadx

dln.3 uaaa

dx

d uu ln.4

The rules to integrating with bases other than e

)23(77ln 2123

xxxx

Differentiate:

ExamplesExamples

x

x

xy

cos

sin

4ln

11

4ln

12

xxy coslog.2 24 xx c o slo glo g 4

24

123

7.3 xxy

xxy 8.4 3

uy u 77ln

xxdx

du23 2 123 xxu

3 22 1(3 2 )(7 ) ln 7x xx x

8ln883 32 xx xxy

xxy 5log.1 27

)5)(7(ln

522 xx

x

xx

xy

5

52

7ln

12

xx

tan4ln

1

4ln

2

4ln

tan

4ln

2 x

x

xx c o slo glo g2 44

xx

tan2

4ln

1

u

u

au

dx

da

ln

1log uaaa

dx

d uu ln

Differentiate:

Practice ProblemsPractice Problems

352 42log.1 xy

)42(2ln

305

4

x

x

42

10

2ln

13

5

4

x

xy

41 0 xu 52 4u x 42log3 52 x

u

u

au

dx

da

ln

1log

)2(2ln

155

4

x

x

Differentiate:

Practice ProblemsPractice Problems

27

2log.2

x

xy

xxy

1

7ln

12

2

1

7ln

1

u

u

au

dx

da

ln

1log

)2)(7(ln

42

)2)(7(ln

xx

x

xx

x

xx )7(ln

2

)2)(7(ln

1

)2)(7(ln

4

xx

x

277 log2log xx xx 77 log22log

)2)(7(ln

42 xx

x

Differentiate:

Practice ProblemsPractice Problems

1. xy

2 sin2. 10x xy

53. 9xy x

xxxx c o s21 01 0ln s i n2

uaaadx

d uu ln

1 0ln1 0c o s2 s i n2 xxxx

9ln995 54 xx xx 54 99ln95 xx xx

xxu c o s2 xxu s in2

lnxx ln

lnx xda a a

dx

5.5 Homework Day 1 AB5.5 Homework Day 1 AB Page 366 1, 7, 19, 21, 27, 37-51 odd, 57

5.5 Homework Day 1 BC5.5 Homework Day 1 BC Page 366 21-55 odds

18

Chapter 5Chapter 5

Transcendental Functions:Transcendental Functions:Bases Other than e

Day 2: Integration examples and Applications.Day 2: Integration examples and Applications.

Find an equation of the tangent line to the graph of .

HWQHWQ

10log 2 5,1y x at

11 5

5 ln10y x

Occasionally, an integrand involves an exponential function to a base other than e. When this occurs, there are two options:(1) use substitution, and then integrate, or (2) integrate directly, using the integration formula

Day 2: IntegrationDay 2: Integration

Example – Integrating an Exponential Function to Another Base

Find ∫2xdx.

Solution:

∫2xdx = + C

Integrate:

dxx 73.5

ExamplesExamples

Cu 55ln

1

2

1

dxx x2

5.6 d xxd u 22xu

duu 52

1C

x

5ln2

52

Cu 33ln

1

7

1

d xd u 7xu 7

duu 37

1 Cx

3ln7

37

Integrate:

Practice ProblemsPractice Problems

dxx 65.7

dxx x 2)3(7)3(.8

Cx

7ln2

72)3(

Cx 2)3(77ln

1

2

1

d xxd u 322)3( xu

duu 56

1 Cx

5ln6

56Cu 5

5ln

1

6

1

d xd u 6xu 6

duu 72

1

Applications of Exponential Functions

Applications of Exponential Applications of Exponential FunctionsFunctions

Suppose P dollars is deposited in an account at an annual interest rate r (in decimal form). If interest accumulates in the account, what is the balance in the account at the end of 1 year? The answer depends on the number of times n the interest is compounded according to the formula

A = P

Applications of Exponential Applications of Exponential FunctionsFunctions

For instance, the result for a deposit of $1000 at 8% interest compounded n times a year is shown in the table.

Applications of Exponential Applications of Exponential FunctionsFunctions

As n increases, the balance A approaches a limit. To develop this limit, use the following theorem.

Applications of Exponential Applications of Exponential FunctionsFunctions

To test the reasonableness of this theorem, try evaluating

it for several values of x, as shown in the table.

Applications of Exponential Applications of Exponential FunctionsFunctions

Now, let’s take another look at the formula for the balance A in an account in which the interest is compounded n times per year. By taking the limit as n approaches infinity, you obtain

Applications of Exponential Applications of Exponential FunctionsFunctions

This limit produces the balance after 1 year of continuous compounding. So, for a deposit of $1000 at 8% interest compounded continuously, the balance at the end of 1 year would be

A = 1000e0.08

≈ $1083.29.

Applications of Exponential Applications of Exponential FunctionsFunctions

Example 6 – Comparing Continuous, Quarterly, and Example 6 – Comparing Continuous, Quarterly, and Monthly CompoundingMonthly Compounding

A deposit of $2500 is made in an account that pays an annual interest rate of 5%. Find the balance in the account at the end of 5 years if the interest is compounded (a) quarterly, (b) monthly, and (c) continuously.

Solution:

Example 6 – Example 6 – SolutionSolution

cont’d

Example 6 – Example 6 – SolutionSolution Figure 5.26 shows how the balance increases over the five-year period. Notice that the scale used in the figure does not graphically distinguish among the three types of exponential growth in (a), (b), and (c).

Figure 5.26

cont’d

5.5 Homework Day 25.5 Homework Day 2 Page 366 59-71 odds, 83, 85