Exponential & Logarithmic Properties

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log(b)b^x

x

b^log(b)x

x

The Product Rule: log(b)(MN)

log(b)N+log(b)M

The Quotient Rule: log(b)(M/N)

log(b)N-log(b)M

The Power Rule: log(b)M^p

plog(b)M

Change of Base (common logs): log(b)M

logM/logb

Change of Base (natural logs): log(b)M

lnM/lnb

The exponential function with base b

f(x)=b^x

The natural exponential function

f(x)=e^x

Formula for interest compounded n times per year

A=P(1+r/n)^(nt)
A= balance
P= prinicpal
t= time in years
n= number of times compounded
r= interest rate (in decimal form)

Formula for interest compounded continuously

A=Pe^(rt)
A= balance
P= prinicpal
t= time in years
r= interest rate (in decimal form)

Express in logarithmic form: y=b^x

log(b)x=y

Express in exponential form: log(b)x=y

y=b^x

e

An irrational number called the natural base
e is about 2.7183

The common log logx means:

log(10)x

The natural log lnx means:

log(e)x

Describe the transformation f(x)=b^(x)+c

Shifts the graph c units upward

Describe the transformation f(x)=b^(x)-c

Shifts the graph c units downward

Describe the transformation f(x)=b^(x+c)

Shifts the graph c units left

Describe the transformation f(x)=b^(x-c)

Shifts the graph c units right

Describe the transformation f(x)=-b^x

Reflects the graph about the x-axis

Describe the transformation f(x)=b^(-x)

Reflects the graph about the y-axis

Describe the transformation f(x)=cb^x

Vertical Stretch c>1
Vertial Shrink 0<c<1
Multiply the y-coordinate of each point by "c"

log(b)N+log(b)M

log(b)(MN)

log(b)N-log(b)M

log(b)(M/N)

plog(b)M

log(b)M^p

Logarithm

the exponent required to produce a given number, this of a positive number y to the base b is defined as follows: If y=b^x, then log b y=x

Asymptote

a line that a graph approaches but never touches

Exponential Function

A function in which the exponent is a variable

Base

watever is being raised to a power

Exponential Growth

When a graph or function changes by increasing amounts

Exponential Decay

When a graph or function changes by decreasing amounts

Logarithm

b ^x=y log y=x
b

Natural Log

e^x=y lny=x

General Form

log-alog (x-h) +k
b

General Form of a Natural Log

aln (x-h) +k

General Form of Exponential

y=a times b^ x-h +k

Simple Interest

Total = Principal + Principal x interest rate x time in years

Interest for one year

Total = Principal x (1 + interest rate)^time in years

Interest for 5 years compounded monthly

Total = Principal x (1 + interest rate/12)^5 x 12

Interest compounded continuously

Total = Principal x e^interest rate x time in years

Population

Find the percent increase or decrease, add or subtract from one, find a when x=0 and write equation - or set up two equations like y=ab^x

Continuously growing populations

N = N0e^kt

Half Life

Find constant (.5N = N0e^k x half-life) Substitute real left over percentage for the .5 to find the years.

Waves

Find scientific notation for large number and take log of number in order to place on number line

Between each level on a logarithmic scale

it increases by a power of 10 (2 levels = 100x)

Finding time difference

take number and raise 10 to that number, divide/subtract it from the other number 10 is raised to - use new number to determine how many times

pH

take negative log of scientific notation

Hydrogen Ion Concentration

take pH and make it the negative exponent of 10, enter into calculator to find scientific notation

Times difference between pH's

take -log of all scientific notations raise 10 to the power of pH's and subtract lower from higher. Raise 10 to the resulting number to find times.

Function

a set of ordered pairs in which no first element repeats

One-to-one Function

a function in which no second element repeats

Exponential Function

a function of the form f(x)=ab×, where the coefficient a≠0, the base b>0 and b≠1

Inverse Function: First Definition

two function f(x) and g(x) are inverse functions of each other if both are one-to-one functions and for every element in their domain f[g(x)]=g[f(x)]=x. The symbol for the inverse function is f(x) is f^-1(x)

Inverse Function: Second Definition

for any one-to-one function f(x), its inverse, f^-1(x), is defined by the following statement: (a,b)is contained in f(x) if and only if (b,a) is contained in f^-1(x)

Inverse Function: Third Definition

The one-to-one functions f(x) and g(x) are each other's inverses if and only if their graphs are symmetric with respect to the diagonal line f(x)=x

y=log(base)bX is by definition equal to

b^y=X where b>0 and b≠1

Common Logarithm

log(base)10X or log X. THEREFORE y=log X is by def'n 10^y=X

logₐ(1/a)

-1

logₐm+logₐn

logₐmn

logₐm-logₐn

logₐ(m/n)

logₐxⁿ

nlogₐx

logₐ(1/x)

-logₐx

aⁿ=b

logₐb=n

logₓa

logₑa/logₑx

logₐaⁿ

n

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)

1/2

2⁻¹

1/9

3⁻²

2

log₆36

Undefined

ln -2

1/125

5⁻³

1/4

8⁻²/³

-1

log₂½

-1

log₄¼

0

log₂/₃1

undefined

ln 0

1/10

100⁻¹/²

2

32¹/⁵

1/5

log₃₂2

1/3

log₈2

-1/3

log₁/₈2

undefined

log 0

log₂mn

log₂m + log₂n

log₇(m/n)

log₇m - log₇n

log₈x⁷

7log₈x

-log₂x

log₂(1/x)

Change the base of log₇x to natural log

ln(x)/ln(7)

percent

a ratio whose denominator is 100
means per hundred

percent change in words

how much a quantity increases or decreases with respect to the original amount

percent change equation

p/100=amount of change/original amount

principal

the original amount of money invested or borrowed

interest

the amount earned or paid for the use of the principal

simple interest

interest is paid only on the principal and only once

compound interest

interest is payed on the principal and previous interest payments
usually compounded at regular intervals

balance

the amount of money in an account at a given time

logarithm (honors)

solves equations for the exponent
an inverse operation to exponents

basic logarithms (honors)

y=b^x
logby=x
read as log base b of x
invented in 1614 by John Napier

change of base property (honors)

logbm=logm/logb
logbm=lnm/lnb

Exponential Function

y=ab^x

Growth Factor

b>1

Decay Factor

b<1

Rate of Change

the percent increase or decrease of an exponential function, (the distance b is away from zero)

Horizontal Asymptote

Imaginary line that acts as the lowest boundary of an exponential function, since it gets very close to zero, but not exactly zero

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