Includes a constant raised to a variable power, f(x) = b^x. The base b must be positive but cannot equal 1
exponential growth
the graph of an exponential function with a base greater than 1
continuous
a smooth curve; there are no gaps in the curve for the domain
horizontal asymptote
a horizontal line that the curve approaches but never reaches
half-life
a fixed period of time in which something repeatedly decreases by half
compounded annually
Interest that builds on itself at 12 month intervals
equivalent equations
All values for x and y that make one equation true also make the other one true ( b^x = b^y if and only if x=y)
one-to-one function
A function that matches each output with one input
logarithmic function
the inverse of an exponential function
inverse function
A function that reverses the effect of another function
product rule for logarithms
states that the logarithm of a product of numbers equals the sum of the logarithms of the factors (log2 4*8 = ?)
quotient rule for logarithms
states that the logarithm of the quotient of two numbers equals the difference of the logarithms of those numbers (log3 81/3 = ?)
power rule of logarithms
states that the logarithm of a power of M can be calculated as the product of the exponent and the logarithm of M (log2 8^16 = ?)
change-of-base formula
State log16 32 as an expression using 2 base logarithms
common logarithm
logarithms with base 10
sound intensity
a measure of how much power sound transmits
sound level
measured in units called decibels (dB); provides a scale that relates how humans perceive sound to a physical measure of its power
irrational constant
The number 'e'. A number that repeats without pattern
natural logarithm
A logarithm with base 'e'
Napierian logarithm
AKA natural logarithm, named after John Napier, a Scottish theologian and mathematician who discovered logarithms
natural base exponential function
a function of form f(x) = ae^rx
continuously compounded interest
interest that builds on itself at every moment f(t) = Pe^rt
Newton's law of cooling
According to this law, the falling temperature obeys an exponential equation (y = ae^cx + T0, where T0 is the temperature surrounding the cooling object, x is the amount of time, and y is the current temperature)
exponential function
a function that can be described by an equation of the form y= a*b^x, where b > 0 and b ≠ 1
exponential growth
as the value of x increases, the value of y increaces
exponential decay
as the value of x increases, the value of y decreases, approaching 0
asymptote
A line that a graph approaches as x or y increases in absolute value
growth factor
for exponential growth y= a*b^x, where b > 1, it is the value of b
decay factor
for exponential decay, 0 < b <1, it is b
exponential growth and decay
A(t) = a(1+r)^t
A is the amount after t time periods
a is the initial amount
r is the rage of growth (r > 0) or decay (r<0)
t is the number of time periods
parent function (exp. fun)
y = b^x
(exp. fun)
y = ab^x
(exp. fun)
y = b^(x-h) + k
all transformations combined (exp. fun)
y = ab^(x-h) + k
continuously compound interest
A=Pe^rt
A is the amount in account at time t
P is the principal
r is the interest rate (annual)
t is time in years
logarithm
this of a positive number y to the base b is defined as follows: If y=b^x, then log b y=x
common logarithm
a logarithm to the base 10
logarithmic scale
every one unit on the scale represents the unit multiplied by ten(1 unit=x10, 2 unit=10x10), when you use the logarithm of a quantity in stead of the quantity you are using the scale.
logarithmic function
the function y=logx that is the inverse of the function y=10^x.
the inverse of an exponential function.
parent functions (log. fun)
y = log↓bx, b > 0, b ≠ 1
(log. fun)
y = a log↓bx
(log. fun)
y = log↓b(x-h) + k
all transformations together (log. fun)
y = a log↓b(x - h) +k
product property
log↓b mn = log↓b m+ log↓b n
quotient property
log↓b m/n = log↓b m - log↓b n
power property
log↓b m^n = n log↓b m
change of base formula
logb M = logc M/ logc ^b, where M, b, and c are positive numbers, and b ≠ 1 and c ≠ 1.
exponential equation
any equation that consists the form b^cx such as a = b^cx where the exponent includes a variable
natural logarithmic function
y=In x or y = log↓ex, the inverse of the natural base exponential
natural logarithmic function
if y = e^x then x = log↓ey = In y. the natural logarithmic function is the inverse of x = In y so you can write it as y = In x
the transformation from logs to exponentials.
General relation
log (base a) of x = y...............means a^y=x
So what the calculator does is find that y number, or the power to which the base a has to be raised so that we get the number x;
Includes a constant raised to a variable power, f(x) = b^x. The base b must be positive but cannot equal 1
exponential growth
the graph of an exponential function with a base greater than 1
continuous
a smooth curve; there are no gaps in the curve for the domain
horizontal asymptote
a horizontal line that the curve approaches but never reaches
half-life
a fixed period of time in which something repeatedly decreases by half
compounded annually
Interest that builds on itself at 12 month intervals
equivalent equations
All values for x and y that make one equation true also make the other one true ( b^x = b^y if and only if x=y)
one-to-one function
A function that matches each output with one input
logarithmic function
the inverse of an exponential function
inverse function
A function that reverses the effect of another function
product rule for logarithms
states that the logarithm of a product of numbers equals the sum of the logarithms of the factors (log2 4*8 = ?)
quotient rule for logarithms
states that the logarithm of the quotient of two numbers equals the difference of the logarithms of those numbers (log3 81/3 = ?)
power rule of logarithms
states that the logarithm of a power of M can be calculated as the product of the exponent and the logarithm of M (log2 8^16 = ?)
change-of-base formula
State log16 32 as an expression using 2 base logarithms
common logarithm
logarithms with base 10
sound intensity
a measure of how much power sound transmits
sound level
measured in units called decibels (dB); provides a scale that relates how humans perceive sound to a physical measure of its power
irrational constant
The number 'e'. A number that repeats without pattern
natural logarithm
A logarithm with base 'e'
Napierian logarithm
AKA natural logarithm, named after John Napier, a Scottish theologian and mathematician who discovered logarithms
natural base exponential function
a function of form f(x) = ae^rx
continuously compounded interest
interest that builds on itself at every moment f(t) = Pe^rt
Newton's law of cooling
According to this law, the falling temperature obeys an exponential equation (y = ae^cx + T0, where T0 is the temperature surrounding the cooling object, x is the amount of time, and y is the current temperature)
exponential function
a function that can be described by an equation of the form y= a*b^x, where b > 0 and b ≠ 1
exponential growth
as the value of x increases, the value of y increaces
exponential decay
as the value of x increases, the value of y decreases, approaching 0
asymptote
A line that a graph approaches as x or y increases in absolute value
growth factor
for exponential growth y= a*b^x, where b > 1, it is the value of b
decay factor
for exponential decay, 0 < b <1, it is b
exponential growth and decay
A(t) = a(1+r)^t
A is the amount after t time periods
a is the initial amount
r is the rage of growth (r > 0) or decay (r<0)
t is the number of time periods
parent function (exp. fun)
y = b^x
(exp. fun)
y = ab^x
(exp. fun)
y = b^(x-h) + k
all transformations combined (exp. fun)
y = ab^(x-h) + k
continuously compound interest
A=Pe^rt
A is the amount in account at time t
P is the principal
r is the interest rate (annual)
t is time in years
logarithm
this of a positive number y to the base b is defined as follows: If y=b^x, then log b y=x
common logarithm
a logarithm to the base 10
logarithmic scale
every one unit on the scale represents the unit multiplied by ten(1 unit=x10, 2 unit=10x10), when you use the logarithm of a quantity in stead of the quantity you are using the scale.
logarithmic function
the function y=logx that is the inverse of the function y=10^x.
the inverse of an exponential function.
parent functions (log. fun)
y = log↓bx, b > 0, b ≠ 1
(log. fun)
y = a log↓bx
(log. fun)
y = log↓b(x-h) + k
all transformations together (log. fun)
y = a log↓b(x - h) +k
product property
log↓b mn = log↓b m+ log↓b n
quotient property
log↓b m/n = log↓b m - log↓b n
power property
log↓b m^n = n log↓b m
change of base formula
logb M = logc M/ logc ^b, where M, b, and c are positive numbers, and b ≠ 1 and c ≠ 1.
exponential equation
any equation that consists the form b^cx such as a = b^cx where the exponent includes a variable
natural logarithmic function
y=In x or y = log↓ex, the inverse of the natural base exponential
natural logarithmic function
if y = e^x then x = log↓ey = In y. the natural logarithmic function is the inverse of x = In y so you can write it as y = In x
the transformation from logs to exponentials.
General relation
log (base a) of x = y...............means a^y=x
So what the calculator does is find that y number, or the power to which the base a has to be raised so that we get the number x;
Studylib tips
Did you forget to review your flashcards?
Try the Chrome extension that turns your New Tab screen into a flashcards viewer!
The idea behind Studylib Extension is that reviewing flashcards will be easier if we distribute all flashcards reviewing into smaller sessions throughout the working day.