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I. Exponential Functions
A. An _____________________ function is a function of the form f(x) = bx, where x is any _________________ number and b > ______ such that b ≠ 1. The constant, b, is called the ___________ of the exponential function.
B. Graphs of exponential functions
1. Since b0 = ______, the graph of f(x) = bx will always contain the point ___________.
2. If b > ______, the graph will look similar to the graph below:
a. the domain is ______________
b. the range is _______________
c. this is a(n) __________________ (increasing or decreasing) function.
d. the line _______________ is a horizontal asymptote
3. If 0 < _____ < 1, the graph will look similar to the graph below:
a. the domain is ______________
b. the range is _______________
c. this is a(n) __________________ (increasing or decreasing) function.
d. the line _______________ is a horizontal asymptote
C. Given the point (2,16) is on the graph of the exponential function f(x) = bx, find f(x).
1. Since the graph contains the point (2, 16), f(2) = _____.
2. Substituting x = 2 and f(2) = 16 into the equation f(x) = bx gives: ______ = b2
3. Taking the square root of both sides gives ±4 = b. Since b > 0 in an exponential function, f(x) = (___)x
D. Graphing exponential functions using transformations
1. The graph below is the graph of y = 3x shifted 2 units to the _________, reflected over the ___________, and shifted 1 unit ___________. [Note the point (0,1) from the original graph is now at (2,0).] It’s new equation is y=-3^(x-2)+1
a. the horizontal asymptote is now the line y = ____
b. the range of the function is now ____________
c. the y intercept can be found by substituting 0 for _____ and solving for ____. So y =-3^(0-2)+1 = _______
E. Solving exponential functions by relating bases
1. The function f(x) = bx is _________________ because it passes the horizontal line test.
2. Solve 1/〖16〗^x =(∛2)^(x+1)
a. Write the left side as (_____)-x and write ∛2 as (_____)1/3 giving
〖16〗^(-x)=(2^(1⁄3) )^(x+1)
b. Write 16 as (____)4, and multiply the exponents on the right side giving
(2^4 )^(-x)=2^((x+1)/3)
c. _________________ the exponents on the left side giving
2^(-4x)=2^((x+1)/3)
d. Equating the exponents gives: ______ = (x+1)/3
e. Solving this equation for x, gives x = -1/13.
F. Compound interest fomula
1. The Compound interest formula is given as A=P(1+r/n)^nt, where ____ is the total amount after t years, P is _______________, r is ____________ rate per year, n is number of times interest is ___________________ per year, and t is the number of years.
II. The Natural Exponential Function
A. Characteristics
1. The function f(x) = (_____)x is called the natural exponential function. The value of e is approximately ____________ (rounded to thousandths).
2. Use a graphing calculator to evaluate:
a. e3.2 = __________
b. 5e-1.72 = __________
B. Solving by relating the bases
1. Solve e^(5x+2)=∛e by first rewriting the equation I the form eu = ev .
x = _________.
C. Applications
1. A = Pert is the ___________________ compound interest formula.
2. A = Pe-rt is the __________________ of A dollars after t years of continuous compound interest.
3. The mathematical model that can describe population growth is gien by the functioin P(t) = P0ekt, where P0 = P(0) is the ______________ population and k is a constant called the relative ____________________.
4. The relative growth rate of a certain bacteria 25%. Suppose there are 1092 bacteria after 16 hours. What was the initial population?
a. Substitute the values we know into the formula. P(16) = ________, k = _____ and t = _____, giving us 1092=P_0 e^(.25(16)). Solving for P0, gives ___________.
III. Logarithmic Functions
A. The inverse function of the ________________ function f(x) = bx, is the ___________________ function f -1(x) = (______)b x
1. y = logb x is said to be in _________________ form and x = by is said to be in ____________________ form.
2. Write 2 = 3x in logarithmic form: _________________
3. Write log5 y = 2 in exponential form:_________________
4. Evaluate log2 32 = ______
B. Properties of Logarithms. For b > 0 and b ≠ 1:
1. logb b = ______
2. logb 1 = ______
3. b^(〖log〗_b x) = ______
4. 〖log〗_b b^x = ______
C. Evaluate the following logarithms using the properties
1. 3^(〖log〗_3 5) = ______
2. log2 1 = _____
3. log2 25 = _____
4. log2 2 = _____
D. Special Bases
1. log x has a base of _____ and is referred to as the ______________ logarithm.
2. ln x has a base of _____ and is referred to as the ______________ logarithm.
3. Evaluate the following logarithms:
a. ln e4 = _____
b. log 1000 = _____
c. ln 1 =
d. ln e = ____
E. Graphing Logarithmic Functions
1. The graph of y = log3 x is the reflection of the graph of y = _____ about the line _________. For every point (x, y) on the graph of y = 3x, the point (_______) is on the graph of y = log3 x.
2. Every logarithmic function of the form y = logb x, where b > 0 and b ≠ 1 has a vertical asymptote at _____________. The domain is ____________ and the range is __________. The graph will always contain the point ___________. The graph is _________________ (increasing, decreasing) if b > 1 and ________________ if 0 < b < 1.
3. The graph of f(x) = -ln(x – 3) – 2 is the graph of y = ln x shifted 3 units ___________, reflected about the ___________, and shifted 2 units ____________. It’s domain is ______________ and its range is _____________. The line _____________ is a vertical asymptote.
4. The domain of a logarithmic function consists of all values of x for which the _________________ of the logarithm is greater than zero. For example, if f(x) = -ln(x – 3) – 2, then __________ > 0. The domain is ____________.
IV. Rules for Logarithms
A. True or False? If b > 0, b ≠ 1, u and v are positive numbers and r is a real number, then:
1. logb uv = logb u + logb v __________
2. (log_b u)/(〖log〗_b v) = logb u – logb v _________
3. 〖log〗_b u/v = logb u – logb v __________
4. (logb u)r = r logb u ___________
5. logb ur = r logb u ___________
B. Write ln ((x-3)^2 (x+2))/(4e^x ) as the sum and difference of logs
1. Use the _____________ rule to write as ln (x-3)2(x+2) – ln 4ex
2. Use the _____________ rule to write as ln (x-3)2 + ln (x+2) – [ ln 4 + ln ex ]
3. Use the _____________ rule and distributive property to write as:
2 ln (x-3) + ln (x+2) – ln 4 – x ln e
4. Finally, use the property that ln e = _____ to write 2 ln (x-3) + ln (x+2) – ln 4 – x
5. In step 3 we could also have used the fact that ln ex = ____.
C. Write 1/4 [logx-3 log(x-1) ]+log8 as a single log
1. Use the _____________ rule to write as 1/4[log〖x-log(x-1)^3]〗+log8
2. Use the _____________ rule to write as 1/4 [log x/(x-1)^3 ]+log8
3. Use the _____________ rule to write as log[x/(x-1)^3 ]^(1/4)+log8
4. Use the _____________ rule to write as log〖8[x/(x-1)^3 ]〗^(1/4)
D. Logarithmic Property of Equality
1. If logb u = logb v, then u = ______
2. If u = v, then logb u = logb(_____)
3. Solve 〖log〗_11 √x=〖log〗_11 6
a. Using the property of equality: √x = _____.
b. Squaring both sides gives x = ______.
E. Change of Base Formula
1. 〖log〗_b u=(〖log〗_a u)/(〖log〗_a b)
2. For example: 〖log〗_3 5= ln5/ln3 = __________
V. Solve Exponential Equations
A. If the bases cannot be related,
1. Use the logarithm property of equality to “take the log of _______________.”
2. Use the power rule of logarithms to “_________________” any exponents.
3. Solve for the given _________________.
B. Solve the equation 3x+7 = 20
1. take the log base 10 ( or base e) of both sides: log 3x+7 = log 20
2. Use the Power rule of logarithms: (________) log 3 = log 20
3. Distribute log 3: x log 3 + 7 log 3 = log 20
4. Subtract ___________ from both sides, then divide by ___________ on both sides to solve or x: x=(log20-7log3)/log3 = _________ (nearest hundredth)
C. It’s better to use the natural logarithm if the base is e. The process will be quicker because ln ex = _____.
1. For example, in solving ex-5 = 32 we take the natural log of both sides to give
ln ex-5 = ln 32
2. The left side simplifies to give us: x – 5 = ln 32
VI. Solve Logarithmic Equations
A. When solving logarithmic equations it is important to always verify the solutions since logarithmic equations often lead to ____________________ solutions.
B. If the equation can be written in the form logb u = logb v, then solve by setting u = ____.
C. If the equation is in the form logb u = a, then eliminate the logarithm by rewriting the equation in __________________ form: ba = u. Then solve for the given variable.
1. Solve log3 (2x + 5) = 2
a. rewrite in exponential form: 32 = ____________
b. simplify and solve for x: 9 = 2x + 5
4 = 2x
2 = x
c. verify the solution: log3 (2*2 + 5) = 2
log3 9 = 2, this is true since 32 = 9
D. If the equation contains more than one log, first combine the logs so that it can be written in the form logb u = a. Then proceed as above
1. Solve log7 (x + 9) + log7 (x + 15) = 1
a. Combine the logs: log7 _________________ = 1
b. Simplify: log7 (x2 + 24x + 135) = 1
c. Write in __________________ form: 71 = x2 + 24x + 135
d. Solve for x: 0 = x2 + 24x + 128
x = ______ or _______
e. Verify the solutions. _______ is an extraneous solution, so ______ is the only solution.
VII. Applications
A. Now that we have the technique to solve exponential equations with the variable in the exponent, we can expand the applications we learned previously
B. Compound interest
1. What is the interest rate necessary for an investment to quadruple after 8 years of continuous compound interest?
a. Using the formula for continuous compound interest, A = Pert.
b. If the initial investment, P, quadruples, then it will be ___ times P. We substitute 8 for ____ and solve for ____.
c. We start with: 4P = Pe8r
d. Dividing both sides by 4 gives ___ = e8r
e. Take the natural log of both sides: ln 4 = _____
f. Dividing by 8 gives r=ln4/8 = ______ = ____%
C. Exponential Decay
1. In an exponential decay model the population exhibits ______________ exponential growth. In other words, the population ____________ over time. The only difference between an exponential growth model and an exponential decay model is that the constant, k, is _______________________ in the decay model.
2. ______________ is the required time for a given quantity to decay to half of its original quantity.
D. Logistic Growth
1. When outside factors exist such as predators or disease that affect the population growth, scientists often us a ______________ model.
2. A model that describes logistic growth is P(t)=C/(1+Be^kt ) where B, C, and k are constants with C > 0 and k < 0. The number C is called the _____________ capacity.
3. The graph of the logistic growth model approaches the horizontal asymptote y = ____.
4. The number of students that hear a rumor on a small college campus t days afer the rumor starts is modeled by the logistic function R(t)=3000/(1+Be^kt ).
a. The carrying capacity for the number of students who will hear the rumor is ______
b. If 8 students initially heard the rumor, find B.
i. R(0) = _____ . So substitute 0 for ____ and 8 for _______
ii. that gives: 8= 3000/(1+Be^0 )=3000/(1+B)
iii. Multiplying both sides by the denominator gives 8 + 8B = 3000, so B = ______
d. If 100 students heard the rumor after 1 day, find k
i. Substituting 374 for ____, 1 for ____, and 100 for ______ gives:
100= 3000/(1+374e^k )
ii. Multiply both sides by the denominator gives 100 + 37400ek = 3000
iii. Subtract 100 and divide by 37400 to give e^k=29/374
iv. Taking the ln of both sides gives k = ln(29/374) ≈ ____________(3 decimal places)
e. How long will it take 2,900 students to hear the rumor?
i. We are looking the time, t, that produces P(t) = 2900
ii. Substituting 2900 for ______, and -2.557 for _____ we now have:
2900= 3000/(1+374e^(-2.557t) )
iii. Multiply by the denominator: 2900 + 1084600e-2.557t = 3000
iv. Solving for e-2.557t we get e-2.557t = 1/10846
v. Solving this exponential equation gives t = ______ days (nearest tenth)
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