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Exponential Function

y = ab^x

exponential growth

increase starts slowly then increases very quickly to infinity

exponential decay

decreasing very quickly, but never reaching zero

Asymptote

An imaginary line on a graph that acts as a boundary line.

Base

the number that is written with an exponent

function that models exponential decay or growth

A - amount after t time periods

a is y intercept

how to identify "a" in exponential growth or decay word problem

growth or decay factor

add annual rate of change to 1

the factor of "a" can

how can affect "a" the graph in exponential growth or decay

y = 2^x

y = 3 * 2^x stretches the graph of the function y = 2^x

continuous compounded interest formula

A - amt in account at time t

What is the irrational number "e"

The number e is a famous irrational number.

2.718281828

What are the first 9 digits of e

Natural base exponential functions

exponential functions with base e

logarithm of a number

logarithm is the exponent to which another fixed value, the base, must be raised to produce that number.

100 = 10^2

use the definition of logarithm

how do you evaluate a logarithm

make the log statement = to x

Common logarithm

A logarithm whose base is 10, or just log

logarithmic scale

using the logarithm of a quantity instead of the quanitity

logarithmic function

the inverse of the exponential function

product property of logarithms

product property of logarithms (can only be used when logarithms have the same base)

quotient property of logarithms

quotient property of logarithms (can only be used when logarithms have the same base)

power property of logarithms

power property of logarithms (can only be used when logarithms have the same base) says that if a log is raised to a power, it can be rewritten as the power times the log

simplifying logarithms

rewrite log expressions as a single logarithm

expanding logarithms

change a single logarithm and expand it to involve the sum or difference of 2 or more logarithms

change of base formula

logby=log base b of y = log y/ log b

exponential equations

any equation that includes a variable with the exponent

Equations

A mathematical statement that contains variables, constant, operations and equal sign.

use

solve logarithmic equations

A(t) = a(1+r)^t

Exponential Growth and Decay Model

Continuously Compounded Interest

A(t) = Pe^rt

Natural logarithmic function

The function f(x)=ln x

growth of decay factor

the value go b in y=ab^x

e^lnx

x

y=a(1+r)ⁿ

What equation expresses exponential growth?

y=a(1-r)ⁿ

What equation expresses exponential decay?

y=a(1+r)ⁿ the "initial amount," or the amount at the beginning.

What does "a" stand for in the equation regarding exponential growth?

y=a(1+r)ⁿ the % increase or decrease

What does "r" stand for in the equation regarding exponential growth?

y=(1±r)

What is the growth/decay factor?

-b/2a (find x and plug it in for y)

How do you find the vertex of a quadratic function?

-b/2a

How do you find the axis of symmetry of a quadratic function?

plug in 0 for x

How do you find the y-int of a quadratic function?

What is a logarithm?

It helps us evaluate exponents that are not whole numbers.

b^x = y

Logby= x if and only if

Log is BAE

Base Answer Exponent

exponent

A mathematical notation indicating the number of times a quantity is multiplied by itself

3

Log(2)^8 = ?

2

Log(5)^25 = ?

0

Log(10)^1 = ?

1

lne = ?

0

ln1 = ?

e

Log(e)e = ?

0

Log(e)1 = ?

-1

ln1/e = ?

e^-1

Log(e)1/e = ?

1/2

What is the sin of 150 degrees?

Log(e)^(radical e) = ?

e^1/2 or (radical e)

5

lne^5 = ?

e^5

Log(e)5 = ?

n • log(b)X

What is another way to write Log(b)x^n

Domain- becomes range

What is true about the domain and range of the original function?

They are reflections over the like y=x

What is true about the graph of a function and its inverse?

Log(b)X

What is the inverse of b^x?

Log(3)X

What is the inverse of 3^x?

a•b^(x-h)+k = f(x)

What is the exponential transformation?

a•log(b)(x-h)+k = f(x)

What is the Log transformation?

What does a represent?

stretch/shrink and a reflection over the x-axis

What does h represent?

shift left or right

What does k represent?

shift up or down

How do we solve exponential and log forms by hand?

find the same base, make the exponents equal to each other, solve for x

exponential growth function (8.1, p465)

expression bⁿ where the base b is a positive number other than 1

graph an exponential function

plug 'n chug (make a table of (x,y) values)

asymptote (8.1, p465)

a line that the graph of the function approaches, but never touches (as you move away from the origin)

exponential growth function (8.1, p466)

y=abⁿ where a > 0, b > 1

Graphing a general exponential function (8.1, p466)

graph y=abⁿ, translate the graph horizontally by h units, and vertically by k units

exponential decay functions (8.2, P474)

f(x) =abⁿ where a>0 and 0<b<1

graph an exponential decay function

plug 'n chug (table of (x,y) pairs)

exponetial decay model (8.2, p476)

get equation a is the initial amount and r is the percent decrease expressed as a decimal the quantity 1-r is called the decay factor.

The natural base e (8.3, p480)

as n approaches ∞, (1+1/n)ⁿ approaches e≈2.7182818

continuously compounded interest (8.3, p482)

pert (get formula) take the compound interest function, and as n aproaches positive infinity it turns into pert which uses e (natural base)

special log values

log of 1 or b⁰=1

graph of log functions (8.4, p489)

- line x-h is a vertical asymtote

change of base formula (8.5, p494)

basically, changing a log expression from arbitrary base c into a log expression of base 10 or a natural log expression.

Solve Exp Equation by equating exponents (8.6, p501)

if 2 powers with the same base are equal, then their exponents must be equal.

Solve Log equation by "equating exp" (8,6, p503)

bases are the same??, then the log "exp" are =