calculus-9/21/2011

Calculus-9/21/2011

Objectives: Solve complex algebraic problems using laws of logs and exponents. Use the definition of log and exponent to switch between log and exponent form in an algebraic equation.

Agenda•Do Now•Laws of Logarithms•Practice•HW: Log Properties Handout

UPCOMING- Test Moved to MONDAY instead of Friday!!

Take Out: Do Now Sheet, Pencil, Homework DO NOW:

Exponent laws handout

Do these on answers out on your green sheet under Wednesday (because guess what it is Wednesday)

Exponents, Logs


Exponents

Exponents are repeated multiplication:

n times

Example:

bbbbbn ...

822223

Objectives: Solve complex algebraic problems using laws of logs and

exponents. Use the definition of log and exponent to switch between log and exponent form in an algebraic

equation.

Rules for Exponents

nn

bb 1

Rule Example

81

212 3

3

10 b 120

mnmn bb

mnmn bbb 633 222

933 22

mnm

n

bbb 43

4

3

222 12

21


Practice

?1 8

?171

?0 33 23

42

2

3

192

)1)2

)3)4

)5

)6

)7Objectives: Solve complex algebraic problems

using laws of logs and exponents. Use the definition of log and exponent to switch between log and exponent form in an algebraic equation.


More Exponent Rules

nba nn ba nvw ba vnwn ba

2

22

ba

ba

nv

nwn

v

w

ba

ba


Practice

332aa1

92

?3

2

3

632ba

)1

)2

)3

)4


Common Mistakes


ROOTS Roots don’t count as a separate category, because they are

just like exponents. We’ll see why in a second.

baba

ba

ba

41682

2428

28


Root – Exponent Connection

nn aa /1 288 33/1

nmm nmn aaa / 3/553 66


Practice

21

16

3/12a2/1

2

4

3 42 gg


Logs

I want you to be able to use logs to solve for a variable.

Things to Remember…If you have an exponential equation with a #

base use logs to solve.If you have an exponential equation with

base e use natural log (ln) to solve.


Logs

Basic Definition of a log:

cab bca log

8log2 3

9log3 2

xgg 3log

3x

3gg x


More Log Rules-Inverse Properties

baa log bb

a alog b

BASE a: BASE e:

xe ln x

xeln x


How can we use this in an algebraic context? Whenever the variable you are looking for is in the

exponent, we need to use logs

43 xe4loglog 3

ex

e e 4lnln 3 xe4ln3 x

3/4lnx


Example 2-using inverse property

100105 y

100log10log 105

10 y

100log5 10y 2

52

y

Example 3- Using Change of Base Rule

1629 z

16log2log9 z

2log16log

2log2log9 z

9449

z

z

Objectives: Solve complex algebraic problems using laws of logs and

exponents. Use the definition of log and exponent to switch between log and

exponent form in an algebraic equation.


Practice-Now you try either use change of base or inverse property to solve for x

1) e2x = 10

2) 54x + 1 = 15

3) 5 ex + 1 = 30

4) ex/5 + 4 = 7

5) 32x = 40


Rules of Logarithms Since a logarithm is simply an exponent which is just

being written down on the line, we expect the logarithm laws to work the same as the rules for exponents, and luckily, they do

Exponents Logarithms

bm × bn = bm+n logb xy = logb x + logb y

bm ÷ bn = bm-n logb (x/y) = logb x − logb y

(bm)n = bmn logb (xn) = n logb x

b1 = b logb (b) = 1

b0 = 1 logb (1) = 0


Example 1 3)3(log)1(log 22 xx

3)3)(1(log 2 xx

)3)(1(23 xx

348 2 xx540 2 xx

1,5 x

Apply product property

Change into exponential form to solve

Simplify

Reduce 1 side to zero to solve the quadraticFactor

Solutions!!

)1)(5(0 xx


Example 2 8)9ln( 4 x

8)9ln(4 x

48)9ln(4 x

29log xe

2)9ln( x

92 xe

9389.7 x

389.16x


Example 3 3)4(log)12(log 22 xx

34)12(log 2

xx

41223

xx

4128

xx

12328 xx

316 x

631

x

Product Property of Logs

Switch into exponential form

Simplify

Get rid of the fraction by multiplying (x-4)

Solve for x

calculus-9/21/2011

Documents

definition of log

algebraic equation

exponent form

complex algebraic problems

laws of logs

exponent laws handout

exponent rules objectives

base use logs