Calculus-9/23/2010
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Transcript of Calculus-9/23/2010
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Calculus-9/23/2010
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-HW Questions-Logs and Exponents powerpointHW: Logs and Exponents Handout
Take Out: Do Now Sheet, Pencil, Homework DO NOW:
Evaluate using laws of exponents
1)
2)
34
x
2
2
24
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Exponents, Logs
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.
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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.
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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
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.
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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.
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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.
More Exponent Rules
nba nn ba nvw ba vnwn ba
2
22
ba
ba
nv
nwn
v
w
ba
ba
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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
332aa1
92
?3
2
3
632ba
)1
)2
)3
)4
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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.
Common Mistakes
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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.
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
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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.
Root – Exponent Connection
nn aa /1 288 33/1
nmm nmn aaa / 3/553 66
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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
21
16
3/12a2/1
2
4
3 42 gg
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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.
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.
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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.
Logs
Basic Definition of a log:
cab bca log
8log2 3
9log3 2
xgg 3log
3x
3gg x
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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.
More Log Rules-Inverse Properties
baa log bb
a alog b
BASE a: BASE e:
xe ln x
xeln x
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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.
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
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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.
Example 2-using inverse property
100105 y
100log10log 105
10 y
100log5 10y 2
52
y
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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.
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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
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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 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
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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.
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
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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.
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
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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.
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