# Exponential & Logarithmic Properties

### 363 terms by plpsopa

#### Study  only

Flashcards Flashcards

Scatter Scatter

Scatter Scatter

## Create a new folder

0

1

x

x

0

1

x

x

0

1

x

x

log(b)N+log(b)M

log(b)N-log(b)M

plog(b)M

logM/logb

lnM/lnb

f(x)=b^x

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)

log(b)x=y

y=b^x

### e

An irrational number called the natural base

log(10)x

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)(MN)

log(b)(M/N)

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

b ^x=y log y=x
b

e^x=y lny=x

### General Form

log-alog (x-h) +k
b

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

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

1

0

-1

logₐmn

logₐ(m/n)

nlogₐx

-logₐx

logₐb=n

logₑa/logₑx

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)

log 10

ln e

2⁻¹

3⁻²

log₃9

½⁻¹

log₆36

ln e⁵

ln -2

64¹/³

5⁻³

8⁻²/³

log₂½

log₄¼

log₆1

log₂/₃1

ln 0

100⁻¹/²

32¹/⁵

log₃₂2

log₈2

log₁/₈2

ln 1

log 0

log₂m + log₂n

log₇m - log₇n

7log₈x

1

0

log₂(1/x)

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

logbm=logm/logb
logbm=lnm/lnb

y=ab^x

b>1

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

See More

Example: