exam 4 Flashcards
12.1
A function that can be written in the form f(x) = b^x where b (1 BLANK) and b (2 BLANK), is called an exponential function with base b.
Two rules for the base of exponential functions.
- Cannot be equal to 1.
- Greater than 0. (b > 0)
12.1
If f(x) = 3^x, find the following:
A. f(3)
B. f(0)
C. f(-2)
A. 27 (3^3)
B. 1 (3^0)
C. 1/9 (3^-2)
12.1
If f(x) = 5 X 3^(x - 4) + 16, find f(6).
Plug it into x. Use a calculator.
61.
12.1
An (BLANK) is a horizontal line that the graph of a function approaches as it moves to the right or left.
Asymptote.
12.1
An (1 BLANK) increasing function is of the form f(x) = b^x where b is greater than (2 BLANK).
- Expontentially
- One
12.1
When graphed, if the base is less than one, the graph is (BLANK).
Think of the oppposite direction.
Decreasing.
12.1
The (BLANK) is denoated with the letter e.
natural base
12.1
What is the natural exponential function?
Base “e”.
f(x) = e^x
12.1
What button accounts for the natural expontential function (the natural base e)?
The “ln” button.
12.1
If f(x) = e^x, find the following rounded to nearest thousandth:
A. f(5)
B. f(-1.7)
B. f(0)
A. 148.413
B. 0.183
C. 1
12.1
For any positive real number b (b isn’t equal to 1) and any real numbers “r” and “s”, if b^r = b^s, then (BLANK).
One-to-one property of exponential functions.
r = s.
12.1
Using one-to-one, solve the following:
A. 2^x = 16
B. 4^x = 256
C. 5^x = 1/125
A. x = 4
B. x = 4
C. x = -3
12.1
Using one-to-one, solve the following:
3^(x + 2) = 1/27
x = -5
12.2
For any positive real number b (b not equal to 1) and any positive real number “x”, we define LOGbx as the (BLANK) that “b” must be raised by to equal “x”.
LOGbx pronounced “the logarithm base b of x” or “log base b of x”
Definition of logarithm.
exponent
12.2
Solve the following logarithmic function:
LOG3of9
What do you raise 3 by to get 9?
9 is the argument.
2.
This is the exponent of the base “3”.
12.2
Solve the following logarithmic function:
LOG5of125
What do you raise 5 by to get 125?
125 is the argument.
3.
This is the exponent of the base “5”.
12.2
For any positive real number “b” (b not equal to 1) and a positive real number “x”, we define f(x) = LOGbx as a logarithmic function.
The function f(x) is defined only for values of “x” greater than 0, so the (BLANK) is (0, infinity).
Think of the two axis on a graph.
domain
12.2
Given the function f(x) = log2ofx, find f(16).
Plug in 16 to x.
4.
12.2
We define the common logarithm whose base is (BLANK).
We write LOGx rather than LOG10ofx.
Common logarithm.
10.
12.2
Evaluate the given logarithm. Round to nearest thousandth.
A. log100
B. log0.001
C. log75
Common logarithm.
A. 2
B. -2
C. 1.875
Remember base of 10.
12.2
The natural logarithm, denoted with lnx is a logarithm whose base is (BLANK).
e
Remember “lnx = LOGeofx”.
12.2
Evaluate the given logarithms. Round to the nearest thousandth.
A. lne^7
B. ln65
C. ln27.3
A. 7
B. 4.174
C. 3.307
12.2
What are the two conversion formulas between exponential form and logarithmic form?
Expontential form:
B^x = Y
Logarithmic form:
LOGBofY = X
“Y” is the argument, “B” is the base, “X” is the exponent
12.2
Rewrite the following in logarithmic form:
A. 2^-8 = 1/156
B. 7^x = 25
C. 3^x = 350
A. LOG2of(1/156) = -8
B. LOG7of(25) = x
C. LOG3of(350) = x
12.2
Rewrite the following in exponential form:
A. LOG3ofX = 6
B. lnx = 7
C. log100 = x
A. 3^6 = x
B. e^7 = x
C. 10^x = 100
12.3
Define the product rule for logarithms.
LOG[base]B[of]X + LOG[base]B[of]Y = LOG[base]B[of]XY
12.3
Define the power rule for logarithms.
LOG[base]b[of]X^r = R * LOG[base]b[of]X
12.3
LOG[base]B[of]B^r = ?
r
12.3
Define the change of base formula below.
LOG[base]B[of]X = ?
LOG[base]A[of]X / LOG[base]A[of]B
OR
ln[of]X / ln[of]B
TIP: Remember you divide the argument of the LOG on top and the base of
12.4
When solving exponential and logarithimic equations, what should you try to do first?
Get the same base (if possible).
12.4
The one-to-one property of logarithimic functions basically says that as long as the LOGS have the same base then…
The arguments are equal. OR The exponents when written in exponential form.
12.5
What is the compound interest formula?
A = P (1 + r/n)^nt
12.5
What does each component represent in the compound interest formula?
A = P (1 + r/n)^nt
P = principle amount
R = interest rate (as decimal)
T = time (in years)
N = number of compounds (per year)
12.5
What is the continuous interest formula?
A = Pe^rt
12.5
What does each component represent in the continuous compound interest formula?
A = Pe^rt
A = amount (after t years)
P = principle
R = interest rate (as a decimal)
T = time (in years)
12.5
What is the exponential growth/decay formula?
Hint: only formula with “p naught.”
P = P(0)e^kt
12.5
What does each component represent in the exponential growth/decay formula?
P = P(0)e^kt
P = size (at time of t)
P(0) = initial size
K = constant
T = time
12.6
When graphing an exponential function, what part of the equation represents the asymptote value?
Hint: think of the standard form of an exponential equation.
The “K” value. It will always affect the y-axis.
12.6
When graphing a logarithmic function, what part of the equation represents the aymptote value?
The “H” value. It will always affect the x-axis.
12.6
What is the standard form of an exponential equation?
f(x) = B^(x-h) - 4
12.6
What is the standard form of a logarithmic equation?
f(x) = LOG[base]B(x-h) + k
12.6
When graphing logarithmic and exponential functions, what can you do first to help find the shift in your graph?
HINT: the first table of points you want to make.
Make (at least) 3 points of the parent graph.
12.6
What do the “H” and the “K” in the standard forms affect in the graph?
The “h” value affects the x-axis and the “k” value affects the y-axis.
