9.1 exponential functions. exponential functions a function of the form y=ab x, where a=0, b>0...
TRANSCRIPT
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9.1 E
XPONENTI
AL
FUNCTI
ONS
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EXPONENTIAL FUNCTIONS
A function of the form y=abx,
where a=0, b>0 and b=1.Characteristics
1. continuous and one-to-one2. domain is the set of all real numbers 3. Range is either all real positive numbers or all real negative numbers depending on whether a is < or > 04. x-axis is a horizontal asymptote5.y-intercept is at a6. y=abx and y=a(1/b)x are reflections across the y-axis
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EXAMPLE 1
Sketch the graph of y=2x. State the domain and range.
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EXAMPLE 2
Sketch y=( )x. State the domain and range.
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EXPONENTIAL GROWTH & DECAY
Exponential Growth:
Exponential function with base greater than one. y=2(3x)
Exponential Decay:
Exponential function with base between 0 and 1 y=4(1/3)x
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EXAMPLE 3-6
Determine if each function is exponential growth or decay
y=(1/5)x y=7(1.2)x
y=2(5)x y=10(4/3)x
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STEPS TO WRITE AN EXPONENTIAL FUNCTION 1. Use the y-intercept to find a
2. Choose a second point on the graph to substitute into the equation for x and y. Solve for b.
3. Write your equation in terms of y=abx (plug in a and b)
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EXAMPLE 7
Write an exponential function using the points (0, 3) and (-1, 6)
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EXAMPLE 8
Write an exponential function using the points (0, -18) and (-2, -2)
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EXAMPLE 9
In 2000, the population of Phoenix was 1,321,045 and it increased to 1,331,391 in 2004.
A. Write an exponential function of the form y=abx that could be used to model the population y of Phoenix. Write the function in terms of x, the number of years since 2000.
B. Suppose the population of Phoenix continues to increase at the same rate. Estimate the population in 2015.
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EXPONENTIAL EQUATIONS
Exponential equation:
An equation in which the variables are exponents
Property of EqualityIf the base is a number other than 1 and the base is the same , then the two exponents equal each other.
2x = 28 then x=8
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STEPS TO SOLVE EXPONENTIAL EQUATIONS/INEQUALITIES1. Rewrite the equation so all terms have like
bases (you may need to use negative exponents)
2. Set the exponents equal to each other
3. Solve
4. Plug x back in to the original equation to make sure the answer works
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EXAMPLE 10
Solve 32n+1 = 81
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EXAMPLE 11
Solve 35x = 92x-1
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EXAMPLE 12
Solve 42x = 8x-1
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EXAMPLE 13
Solve 256
14 13 p
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EXAMPLE 14
Solve 1255 32 x
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EXAMPLE 15
Solveaa 164 64
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9.2 LO
GARITHMS A
ND
LOGARIT
HMIC F
UNCTIONS
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Logarithms with base b
Say: “Log base b of x equals y.”
yxb log
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LOGARITHMIC TO EXPONENTIAL FORM
216log.2 4
327
1log.3
01log.1
3
8
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EXPONENTIAL TO LOGARITHMIC FORM
39.6
1000.4
2
1
3
a 322.5 5
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EVALUATE LOGARITHMIC EXPRESSIONS
64log.7 2 81log.8 3
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CHARACTERISTICS OF LOGARITHMIC FUNCTIONS
1. Inverse of the exponential function y=bx
2.Continous and one-to-one
3. Domain is all positive real numbers and range is ARN
4. y-axis is an asymptote
5. Contains (1,0), so x-intercept is 1
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HELPFUL HINT
Since exponential and logarithmic functions are inverses if the bases are the same they “undo” each other…
143
86log
)14(log
86
3
xx
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LOGARITHMIC EQUATIONS
Property of Equality If b is a positive number other than 1, then if and only if x = y.
yx bb loglog
3
3loglog 77
x
x
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EXAMPLE 9
Solve2
5log4 n
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EXAMPLE 10
Solve )34(loglog 42
4 xx
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EXAMPLE 11
Solve pp 5
25 log)2(log
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LOGARITHMIC TO EXPONENTIAL INEQUALITY
3
2
2
3log
x
x5
3
30
5log
x
x
If b > 1, x > 0 and logbx > y then x > by
If b > 1, x > 0 and logbx < y then 0< x < by
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EXAMPLE 12
Solve 2log5 x
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EXAMPLE 13
Solve 3log4 x
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PROPERTY OF INEQUALITY FOR LOGARITHMIC FUNCTIONS
If b>1, then if and only if x>y
and if and only if x<y
yx bb loglog
yx bb loglog
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EXAMPLE 14
)6(log)43(log 1010 xx
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EXAMPLE 15
)5(log)82(log 77 xx
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9.3 P
ROPERT
IES O
F
LOGARIT
HMS
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PRODUCT PROPERTY
The logarithm of a product is the sum of the logarithm of its factors
nmnm bbb loglog))((log
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QUOTIENT PROPERTY
The logarithm of a quotient is the difference of the logarithms of the numerator and denominator.
nmn
mbbb logloglog
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POWER PROPERTY
The logarithm of a power is the product of the logarithm and the exponent
mpm bp
b loglog
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EXAMPLE 1
16log4loglog3 555 x
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EXAMPLE 2
2)6(loglog 44 xx
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EXAMPLE 3
3log27loglog2 777 x
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EXAMPLE 4
125log5loglog4 222 x
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EXAMPLE 5
7loglog42log 333 n
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EXAMPLE 6
9loglog2 55 x
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9.4 C
OMMON
LOGARIT
HMS
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COMMON LOGARITHMS
Logarithms with base 10 are common logs
You do not need to write the 10 it is understood
Button on calculator for common logs
100logLOG
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EXAMPLES: USE CALCULATOR TO EVALUATE EACH LOG TO FOUR DECIMAL PLACES
1. log 3 2. log 0.2
3. log 5 4. log 0.5
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SOLVE LOGARITHMIC EQUATIONSExample 5:
The amount of energy E, in ergs, that an earthquake releases is related to is Richter scale magnitude M by the equation logE = 11.8 + 1.5M. The Chilean earthquake of 1960 measured 8.5 on the Richter scale. How much energy was released?
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Example 6:
Find the energy released by the 2004 Sumatran earthquake, which measured 9.0 on the Richter scale and led to the tsunami.
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HELPFUL HINT
If both sides of the equation cannot be easily written as powers of the same base you can solve by taking the log of each side!
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EXAMPLE
3x=11 4x=15
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SOLVING INEQUALITIES
Example 7
53y<8y-1
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EXAMPLE 8
32x>6x+1
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EXAMPLE 9
4y<52y+1
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CHANGE OF BASE FORMULA
5log
12log12log
10
105
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EXAMPLE
Express in terms of common logs, and then approximate its value to four decimal places.
log425 log318 log7 5
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9.5 B
ASE E A
ND NAT
URAL
LOGS
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NATURAL BASE EXPONENTIAL FUNCTION
An exponential function with base e e is the irrational number 2.71828…
*These are used extensively in science to model quantities that grow and decay continuously
Calculator button ex
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EVALUATE TO FOUR DECIMAL PLACES
1. e2 2. e-1.3 3. e1/2
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THE LOG WITH BASE E IS A NATURAL LOGWritten as : ln
y=ln x is the inverse of y = ex
All properties for logs apply the same way to natural logs
Calculator button lnx
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EXAMPLES
Use calculator to evaluate to four decimal places
4. ln4 5. ln0.056. ln7
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EXAMPLE
Write an equivalent exponential or log equation to the given equation.
7. ex=5 8. lnx≈0.6931
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REMEMBER…..
All log properties apply to natural logs
Do the same thing for ln problems that you do for log problems
Let’s solve!!!!!!!!!
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EXAMPLE 9
Solve e4x=120 and round to four decimal places
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EXAMPLE 10 EXAMPLE 11
ex-2 + 4<21 ln6x > 4
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EXAMPLE 12 EXAMPLE 13
ln5x+ln3x>9 2e3x + 5 =2
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9.6 E
XPONENTI
AL
GROWTH
AND D
ECAY
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EQUATIONS THAT DEAL WITH E
Continuously Compounded InterestA=Pert
A= amount in account after t yearst= # of yearsr= annual interest rateP= amount of principal invested
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EXAMPLES
Suppose you deposit $1000 in an account paying 2.5% annual interest, compounded continuously.
Find the balance after 10 years
Find how long it will take for the balance to reach at least $1500
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Suppose you deposit $5000 in an account paying 3% annual interst, compounded continuously.
Find what the balance would be after 5 years
Find how long it will take for the balance to reach at least $7000
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EXPONENTIAL DECAY
y=a(1-r)t
a=initial amount r=% of decrease expressed as a decimal, this is also called
rate of decay t=time
y=ae-kt
a=initial amount k=constant t=time
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EXAMPLE 3
A cup of coffee contains 130 milligrams of caffeine. If caffeine is eliminated from the body at a rate of 11% per hour, how long will it take for half of this caffeine to be eliminated?
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EXAMPLE 4
The half-life of Sodium-22 is 2.6 years. What is the value of k and the equation of decay for
Sodium-22?
A geologist examining a meteorite estimates that it contains only about 10% as much Sodium-22 as it would have contained when it reached Earth’s surface. How long ago did the meteorite reach Earth?
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EXPONENTIAL GROWTH
y=a(1+r)t
a= initial amount r=% of increase/growth expressed as a decimal t=time
y=aekt
a=initial amount k=constant t=time
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EXAMPLE 5
Home values in Millersport increase about 4% per year. Mr. Thomas purchased his home eight years ago for $122,000. What is the value of his home now?
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EXAMPLE 6
The population of a city of one million is increasing at a rate of 3% per year. If the population continues to grow at this rate, in how many years will the population have doubled?
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EXAMPLE 7
Two different types of bacteria in two different cultures reproduce exponentially. The first type can be modeled by B1(t)=1200e0.1532t and the second can be modeled B2(t)=3000e0.0466t where t is the number of hours. According to these models, how many hours will it take for the amount of B1 to exceed the amount of B2?