Download x – 3

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7.5 - Exp and Log Equations and Inequalities
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
Product:

Quotient:

Power:
a a a
n
m
n
a

a
m
a
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n–m
(a )  a
n m
n+m
n* m
7.5 - Exp and Log Equations and Inequalities
2

Product: logb ( x)  logb ( y)  logb ( xy)
x
 Quotient: log b x  log b y  log b  
 y

Power:
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log b (a )  p log b ( a)
p
7.5 - Exp and Log Equations and Inequalities
3
If each equation on both sides are exponents:
1. Rewrite both sides by “log”-ing it
2. Use exponent and/or logarithmic
rules
3. Solve algebraically, Round to 4
decimal places
4. Check
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7.5 - Exp and Log Equations and Inequalities
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Solve and Check: 98 – x = 27x – 3
log 9 8 – x = log 27 x – 3
Rewrite both sides by “log”-ing it
(8 – x) log 9 = (x – 3) log 27
log 9
log 27
(8  x)
 ( x  3)
log 9
log 9
8 – x = (x – 3) (1.5)
8 – x = (1.5)x – 3
8 – x = 1.5x – 4.5
x=5
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Follow the log rules;
Power Rules
Use Algebra to solve
Distribute 1.5 to x - 3
Solve for x.
Answer.
7.5 - Exp and Log Equations and Inequalities
5
Solve and Check: 98 – x = 27x – 3
9 8 – x = 27 x – 3
2(8  x )
3
3( x 3)
3
16 – 2x = 3x –9
x=5
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Since 27 is a base of 3, apply it to both sides
Use Algebra to solve
Solve for x
Answer.
7.5 - Exp and Log Equations and Inequalities
6
Solve and Check: 8 x = 2 x + 6
x=3
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7.5 - Exp and Log Equations and Inequalities
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Solve and Check: 43x–1 = 8x+1
x = 5/3
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7.5 - Exp and Log Equations and Inequalities
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Solve and Check: 4x – 1 = 5
log 4 x – 1 = log 5
Rewrite both sides by “log”-ing it
(x – 1) log 4 = log 5
Follow the log rules;
Power Rules
log 4 log 5
( x  1)

log 4 log 4
Use Algebra to solve
x – 1 ≈ 1.1610
x ≈ 2.1610
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Solve for x; Round to four decimal places
Answer.
7.5 - Exp and Log Equations and Inequalities
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Solve and Check: 32x–1 = 20
x ≈ 1.8634
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7.5 - Exp and Log Equations and Inequalities
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If each equation on one side shows a log.:
1a. Rewrite the equation in
exponential form
1b. Use exponent and/or logarithmic
rules (including Change of Base)
2. Solve algebraically, Round to 4
decimal places
3. Check
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7.5 - Exp and Log Equations and Inequalities
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Solve : log7(5x + 3) = 3
3
7 = 5x + 3
343 = 5x + 3
5x = 340
x = 68
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Rewriting the equation in exponential form
Use Algebra to solve for x
Solve for x.
Answer.
7.5 - Exp and Log Equations and Inequalities
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Solve : log6(2x – 1) = –1
x = 7/12
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7.5 - Exp and Log Equations and Inequalities
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Solve : log4100 – log4(x + 1) = 1
Can this equation be written in Exponential Form?
 100 
log 4 
 1
 x 1 
100
1
4 
x 1
100
4
x 1
4( x  1)  100
x = 24
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NO
Write problem using Log properties
Rewrite equation using exponential form
to solve
Solve for x.; cross multiply
Answer.
7.5 - Exp and Log Equations and Inequalities
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Solve : log12x + log12(x + 1) = 1
Why can’t
x = –4?
---------------------Plug –4 into
original equation.
----------------------
x=3
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7.5 - Exp and Log Equations and Inequalities
The answer is
undefined.
15
Solve for x:
1. 23x = 15
2. 7–x = 21
3. log5x 4 = 8
4. 3 = log 8 + 3log x
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7.5 - Exp and Log Equations and Inequalities
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Suppose a bacteria culture doubles in size
every hour. How many hours will it take for the
number of bacteria to exceed 1,000,000?
At hour 0, there is one bacterium, or 20 bacteria. At
hour one, there are two bacteria, or 21 bacteria, and so
on. So, at hour n there will be 2n bacteria.
Solve 2n > 106
Write 1,000,000 in scientific
annotation.
log 2n > log 106
Take the log of both sides.
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7.5 - Exp and Log Equations and Inequalities
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Suppose a bacteria culture doubles in size every hour. How many hours will
it take for the number of bacteria to exceed 1,000,000?
nlog 2 > log 106
Use the Power of Logarithms.
nlog 2 > 6
log 106 is 6.
6
log 2
6
n>
0.301
Divide both sides by log 2.
n > ≈ 19.94
Round up to the next whole number.
n>
Evaluate by using a calculator.
It will take about 20 hours for the number of bacteria to
exceed 1,000,000.
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7.5 - Exp and Log Equations and Inequalities
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Worksheet
Pg 526 8, 21 – 33 all NOT 27
Non-Calc Quiz Friday
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7.5 - Exp and Log Equations and Inequalities
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