exponential function

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;