bruce mayer, pe licensed electrical & mechanical engineer bmayer@chabotcollege
DESCRIPTION
Chabot Mathematics. §9.5a Exponential Eqns. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected]. MTH 55. 9.4. Review §. Any QUESTIONS About §9.4 → Logarithm Change-of-Base Any QUESTIONS About HomeWork §9.4 → HW-47. Summary of Log Rules. - PowerPoint PPT PresentationTRANSCRIPT
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Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
§9.5a§9.5aExponential EqnsExponential Eqns
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Bruce Mayer, PE Chabot College Mathematics
Review §Review §
Any QUESTIONS About• §9.4 → Logarithm Change-of-Base
Any QUESTIONS About HomeWork• §9.4 → HW-47
9.4 MTH 55
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Bruce Mayer, PE Chabot College Mathematics
Summary of Log RulesSummary of Log Rules
For any positive numbers M, N, and a with a ≠ 1
log log log ;a a aM
M NN
log log ;pa aM p M
log .ka a k
log ( ) log log ;a a aMN M N
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Bruce Mayer, PE Chabot College Mathematics
Typical Log-ConfusionTypical Log-Confusion
BewareBeware that Logs do NOT behave Algebraically. In General:
loglog ,
loga
aa
MM
N N
log ( ) (log )(log ),a a aMN M N
log ( ) log log ,a a aM N M N
log ( ) log log .a a aM N M N
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Bruce Mayer, PE Chabot College Mathematics
Solving Exponential EquationsSolving Exponential Equations
Equations with variables in exponents, such as 3x = 5 and 73x = 90 are called EXPONENTIAL EQUATIONS
Certain exponential equations can be solved by using the principle of exponential equality
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Bruce Mayer, PE Chabot College Mathematics
Principle of Exponential EqualityPrinciple of Exponential Equality
For any real number b, with b ≠ −1, 0, or 1, then
bx = by is equivalent to x = y
That is, Powers of the same base are equal if and only if the exponents are equal
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Bruce Mayer, PE Chabot College Mathematics
Example Example Exponential Equality Exponential Equality
Solve for x: 5x = 125 SOLUTION Note that 125 = 53. Thus we can write
each side as a power of the same base:
5x = 53
Since the base is the same, 5, the exponents must be equal. Thus, x must be 3. The solution is 3.
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Bruce Mayer, PE Chabot College Mathematics
Example Example Exponential Equality Exponential Equality
Solve each Exponential Equationa. 25x 125 b. 9x 3x1
SOLUTION
a. 52 x53
52 x 53
2x 3
x 3
2
b. 32 x3x1
32 x 3x1
2x x 1
2x x 1
x 1
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Bruce Mayer, PE Chabot College Mathematics
Principle of Logarithmic EqualityPrinciple of Logarithmic Equality
For any logarithmic base a, and for x, y > 0,
x = y is equivalent to logax = logay
That is, two expressions are equal if and only if the logarithms of those expressions are equal
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Bruce Mayer, PE Chabot College Mathematics
Example Example Logarithmic Equality Logarithmic Equality
Solve for x: 3x+1 = 43 SOLUTION
3 x +1 = 43
log 3 x +1 = log 43
(x +1)log 3 = log 43
x +1 = log 43/log 3
x = (log 43/log 3) – 1 2.4236.x
Principle of logarithmic equality
Power rule for logs
The solution is (log 43/log 3) − 1, or approximately 2.4236.
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Bruce Mayer, PE Chabot College Mathematics
Example Example Logarithmic Equality Logarithmic Equality
Solve for t: e1.32t = 2000 SOLUTION
5.7583.t
Note that we use the natural logarithm
Logarithmic and exponential functions are inverses of each other
e1.32t = 2000
ln e1.32t = ln 2000
1.32t = ln 2000
t = (ln 2000)/1.32
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Bruce Mayer, PE Chabot College Mathematics
To Solve an Equation of the To Solve an Equation of the Form Form aatt = = bb for for tt
1. Take the logarithm (either natural or common) of both sides.
2. Use the power rule for exponents so that the variable is no longer written as an exponent.
3. Divide both sides by the coefficient of the variable to isolate the variable.
4. If appropriate, use a calculator to find an approximate solution in decimal form.
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Bruce Mayer, PE Chabot College Mathematics
Example Example Solve by Taking Logs Solve by Taking Logs
Solve each equation and approximate the results to three decimal places.
a. 2x 15 b. 52x 2 17
SOLUTION a. 2x 15
ln 2x ln15
x ln 2 ln15
x ln15
ln 23.907
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Bruce Mayer, PE Chabot College Mathematics
Example Example Solve by Taking Logs Solve by Taking Logs
SOLUTION
b. 52x 3 17
2x 3 17
5
ln 2x 3 ln17
5
x 3 ln 2 ln17
5
x 3 ln
175
ln 2
x ln
175
ln 2 3
x 4.766
b. 52x 2 17
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Bruce Mayer, PE Chabot College Mathematics
Example Example Different Bases Different Bases
Solve the equation 52x−3 = 3x+1 and approximate the answer to 3 decimals
SOLUTION ln 52 x 3 ln 3x1
2x 3 ln 5 x 1 ln 3
2x ln 5 3ln 5 x ln 3 ln 3
2x ln 5 x ln 3 ln 3 3ln 5
x 2 ln 5 ln 3 ln 3 3ln 5
x ln 3 3ln 5
2 ln 5 ln 32.795
Take ln of both sides
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Bruce Mayer, PE Chabot College Mathematics
Example Example Eqn Quadratic in Form Eqn Quadratic in Form
Solve for x: 3x − 8∙3−x = 2. SOLUTION 3x 3x 83 x 2 3x
32 x 830 23x
32 x 8 23x
32 x 23x 8 0
This equation is quadratic in form. Let y = 3x then y2 = (3x)2 = 32x. Then,
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Bruce Mayer, PE Chabot College Mathematics
Example Example Eqn Quadratic in Form Eqn Quadratic in Form
Solncont.
32 x 23x 8 0
y2 2y 8 0
y 2 y 4 0
y 2 0 or y 4 0
y 2 or y 4
3x 2 or 3x 4 But 3x = −2 is not possible because
3x > 0 for all numbers x. So, solve 3x = 4 to find the solution
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Bruce Mayer, PE Chabot College Mathematics
Example Example Eqn Quadratic in Form Eqn Quadratic in Form
Solncont.
3x 4
ln 3x ln 4
x ln 3 ln 4
x ln 4
ln 3x 1.262
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Bruce Mayer, PE Chabot College Mathematics
Example Example Population Growth Population Growth
The following table shows the approximate population and annual growth rate of the United States of America and Pakistan in 2005
CountryPopulatio
n
Annual Population
Growth Rate
USA 295 million 1.0%
Pakistan 162 million 3.1%
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Bruce Mayer, PE Chabot College Mathematics
Example Example Population Growth Population Growth
Use the population model P = P0(1 + r)t and the information in the table, and assume that the growth rate for each country stays the same.
In this model, • P0 is the initial population,
• r is the annual growth rate as a decimal
• t is the time in years since 2005
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Bruce Mayer, PE Chabot College Mathematics
Example Example Population Growth Population Growth
Use P = P0(1 + r)t and the data table:
a. to estimate the population of each country in 2015.
b. If the current growth rate continues, in what year will the population of the United States be 350 million?
c. If the current growth rate continues, in what year will the population of Pakistan be the same as the population of the United States?
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Bruce Mayer, PE Chabot College Mathematics
Example Example Population Growth Population Growth
SOLUTION: Use model P = P0(1 + r)t
a. US population in 2005 is P0 = 295. The year 2015 is 10 years from 2005.
P 295 1 0.01 10 325.86 million
Pakistan in 2005 is P0 = 162
P 162 1 0.31 10 219.84 million
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Bruce Mayer, PE Chabot College Mathematics
Example Example Population Growth Population Growth
SOLUTION b.: Solve for t to find when the United States population will be 350.
350 295 1 0.01 t
350
295 1.01 t
ln350
295
ln 1.01 t
ln350
295
t ln 1.01
t ln
350295
ln 1.01 17.18
Some time in yr 2022 (2005 + 17.18) the USA population will be 350 Million
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Bruce Mayer, PE Chabot College Mathematics
Example Example Population Growth Population Growth
SOLUTION c.: Solve for t to find when the population will be the same in both countries. 295 1 0.01 t 162 1 0.031 t
295 1.01 t 162 1.031 t
295
162
1.031
1.01
t
ln295
162
ln
1.031
1.01
t
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Bruce Mayer, PE Chabot College Mathematics
Example Example Population Growth Population Growth
Soln c.cont. ln
295
162
t ln
1.031
1.01
t ln
295162
ln1.0311.01
29.13
Some time year 2034 (2005 + 29.13) the two populations will be the same.
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Bruce Mayer, PE Chabot College Mathematics
WhiteBoard WorkWhiteBoard Work
Problems From §9.5 Exercise Set• 16, 20, 32, 34, 36, 40
logistic difference equation by Belgian ScientistPierre Francois Verhulst
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Bruce Mayer, PE Chabot College Mathematics
All Done for TodayAll Done for Today
EMP WidmarkBAC Eqn
Calculator
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Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
AppendiAppendixx
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srsrsr 22