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Exponential & Logarithmic Properties

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log(b)1
0
log(b)b
1
log(b)b^x
x
b^log(b)x
x
log1
0
log10
1
log10^x
x
10^logx
x
ln1
0
lne
1
lne^x
x
e^lnx
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
logₐa
1
logₐ1
0
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
log 10
1
ln e
1/2
2⁻¹
1/9
3⁻²
2
log₃9
2
½⁻¹
2
log₆36
5
ln e⁵
Undefined
ln -2
4
64¹/³
1/125
5⁻³
1/4
8⁻²/³
-1
log₂½
-1
log₄¼
0
log₆1
0
log₂/₃1
undefined
ln 0
27
1/10
100⁻¹/²
2
32¹/⁵
1/5
log₃₂2
1/3
log₈2
-1/3
log₁/₈2
0
ln 1
undefined
log 0
log₂mn
log₂m + log₂n
log₇(m/n)
log₇m - log₇n
log₈x⁷
7log₈x
log₂2
1
log₅1
0
-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
"a" in the exponential function equation
starting value
"b" in the exponential function equation
rate of change
Compound Interest Formula
A=P(1+(r/n))^(n/t)
"P" in the compound interest formula
starting value
"r" in the compound interest formula
interest rate
"n" in the compound interest formula
number of times compounded per year
"t" in the compound interest formula
time in years
continuously compounding interest formula
A=Pe^rt
"P" in the continuously compounding interest formula
starting value
"r" in the continuously compounding interest formula
interest rate
"t" in the continuously compounding interest formula
time in years
Half-life Formula
y=a(.5)^(t/h)
"a" in the half-life formula
starting value
"t" in the half-life formula
time of measurement
"h" in the half-life formula
half-life
exponential form of logarithms
y=b^x
logarithm form
logb^y=x
log. loop
b raised to the x = y
common logarithm
when there is no given base, assume it is 10
power property of logs
LOGbM^k to kLOGbM
quotient property of logs
LOGb(M/N) to LOGbM-LOGbN
product property of logs
LOGb(MN) to LOGbM+LOGbN
exponential equations with variables equation
a=b^cx
solving exponential equations w/ variables
-move the exponent in front of the log by the power rule
-take the log of both sides to get x by istelf
change of base formula
LOGbM to LOG10M / LOG10b
increases toward infinity
In a standard exponential function y = 2^x, when x increases toward infinity, y ___.
increases toward zero
In a standard exponential function y = 2^x, when x decreases towards negative infinity, y ___.
e
number with a value of 2.718
exponential function
y = a^x
c units to the right
f(x) = a^(x-c) shifts f(x) = a^x ___
c units to the left
f(x) = a^(x+c) shifts f(x) = a^x ___
c units up
f(x) = a^(x) + c shifts f(x) = a^x ___
c units down
f(x) = a^(x) - c shifts f(x) = a^x ___
y = c
the horizontal asymptote of f(x) = a^(x) +/- c
flip over x-axis
f(x) = -a^x will make f(x) = a^x ___
flip over y-axis
f(x) = a^(-x) shifts f(x) = a^x ___
bases are the same
you can solve exponential equation only if the ____
logarithms
exponents
exponential function
___ is the inverse of a logarithm
exponential function
y = 2^x
logarithmic functions
x = 2^y
b
base value
e
exponent
a
answer
positive
the b and a values must be ___
positive or negative
the e value can be ___
1
the b value cannot equal ___
x
log v b B^x
x
b^(log v b (x))
log v b m + log v b n
log v b mn
log v b m - log v b n
log v b (m/n)
(x) log v b m
log v b m^x
Logarithm - Definition
The inverse of taking the exponent of something
log₃27 =3
3³ =27
log₈1/64 = -2
8⁻² = 1/64
Logarithm Property 1
log₂8 + log₂32 = log₂256
Logarithm Property 2
log₃1/9 - log₃81 = log₃1/729
Logarithm Property 3
3 ∗ log₂8 = log₂ of 8³
Logarithm Property 4
log₁₇357 = log₁₀357/log₁₀17
Logarithm Property 5
= log₂√32/√8
= log₂(32/√8)¹/²
=1/2 ∗ log₂(32/√8)
= 1/2(log₂32 -1/2 ∗ log₂8)
= log₂32 - 1/4 ∗ log₂8
= 5/2 - 3/4 = 7/4
LogxAB=logxA+logxB
EX:
Log₄192=3.7925
Product Property
LogxA/B=LogxA-LogxB
Quotient Property
LogbMp=PlogGbM
EX:log₂25=4.6438
Power Property
if logxA=logxB, than a=b
EX:log₆x+log₆(x-9)=2
log₆x(x-9)=2
x(x-9)=6²
x²-9x-36=0
(x-12)(x+3)=0
x-12=0 or x+3=0
x=12 x=-3
Check which anwser works in the problem
in the calculator.
Equality Property
Log10x=Logx
SUPER EASY DO IN CALCULATOR
Common Logarithms
LogxY=logY/LogX
EX.
4^x=19
log4^X=log19
xlog4=log19
x=log19/log4
x≈2.1240
Change of base formula
A=PErt
A=Amount after T years
P=Principal (original) amount intrested
R=Annual interest rate
EX.
A=3000e^(0.04)(10)
A=3000e^0.4
A≈$4475.47
DON'T FORGET TO PUT THE DECIMAL PLACES INTO THE HUNDREDTH PLACE FOR MONEY!!!!
Compound Interest Formula
4e^-2x -5=3
4e^-2x=8
e^-2x=2
ln e^-2x=ln 2
-2x=ln 2
x=ln2/-2
x≈-0.3466
...
TRY THIS:
3e^4x -12=15
Base E
For any positive number u, except 1
y = ᵤᶢ → logᵤy = ᶢ
When y > 0, When u > 0, u ≠ 1
logᵤy = Defined
Defined
logᵤu = 1
Defined
logᵤ1 = 0
If u and x are positive, a ≠ 1
logᵤxʳ = rlogᵤx
If u,x,y are positive, a ≠ 1
logᵤxy = logᵤx + logᵤy
If u,x,y are positive, a ≠ 1
logᵤ(x/y) = logᵤx - logᵤy
For any number r
r = rlogᵤu = logᵤuʳ
If i,r,u are positive, i ≠ 1, u ≠ 1
logᵢr = logᵤr/logᵤi
If i,r,u are positive, i ≠ 1, u ≠ 1
logᵢr = 1/logᵣi
exponential function
A function with a variable in the exponent. Have horizontal asymptotes.
logarithmic function
The inverse of an exponential function. Have vertical asymptotes.
property of equality
you use this when solving an equation: you must do the same thing to both sides of an equation
expression
a group of symbols that make a mathematical statement
equation
A mathematical sentence that contains an equals sign.
argument of a log
must be greater than zero
common logarithms
logarithms with base 10
natural logs
logarithms with base e
The natural exponential function
f(x) = e^x
The Euler number
approximately 2.718
example of polynomial function
f(x)=3x^4 - 2x^3 + 3x -5
rational function
a function that is expressed as the quotient of 2 polynomial functions.
T-chart for graphing log functions
1,0, base, 1
T-chart for graphing exponential functions
0, 1, 1, base
change to exponential form: log 100 = 2
10^2 = 100
change to log form: e^0 = 1
ln 1 = 0
log 1 =
0
ln e =
1
ln 1
1
anything raised to the zero power is what?
1
when you multiply like bases you do what to the exponents?
add
when you divide like bases you do what to the exponents?
subtract
when you add like terms you do what to the exponent?
nothing
exponential growth
the base is an integer and greater than 1 with no reflection.
exponential decay
has a base between 0 and 1 (a fraction) with no reflection.
exponential growth
may have a base between 0 and 1 if there IS a reflection
a square root
is a power of 1/2
the pivot point
is the top numbers in the T-chart after transformations
to find the y-intercept for exponential functions
put 0 in for x and solve
compounding continuously
A=Pe^rt
compounding a specific # of times per year
A=P(1+r/n)^nt
doubling
A is 2 and P is 1
half life
A is .5 and P is 1
decay
A = pe^(-kt)
calculator input
put parenthesis around the entire denominator and around the entire power.
Logarithm of X
power to which a base number must be raised to equal X
log
base-10
ln
base-e
log (mn)
log m + log n
log (m/n)
log m - log n
log (m^n)
(n) log m
log(b)1
0
log(b)b
1
log(b)b^x
x
b^log(b)x
x
log1
0
log10
1
log10^x
x
10^logx
x
ln1
0
lne
1
lne^x
x
e^lnx
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
what does n^5 mean?
That n is multiplied by itself 4 times (n x n x n x n x n); note this is not 5 because n^1 is multiplied x 1, n^2 is multiplied by itself once (n x n), etc.
a^m x a^n =
a^ (m+n) // to multiply numbers with the same base, keep the base and ADD the indices -p. 386
a^m / a^n =
a^(m-n) // page 386
(a^m)^n =
a^(mxn) // page 386
(ab)^n =
a^n x b^n // the power of a product is the product of the powers; p. 387
(a/b)^n
a^n / b^n -387
a^0 =
1 as long as a is not zero in which case it's zero
a^(-1) =
1/a // remember that an exponent changes its sign when you move from numerator to denominator and back again -387
a^(1/n) =
nth root of a // see p. 389 for the notation; note that the square root of a number is the same as that number raised to the power of 1/2.
square root of a can also be expressed as a to the power of ...
a^(1/2) -389
What is 8^(2/3) [8 to the power of two-thirds]?
8^(2/3) = (8^(1/3))^2 = 2^2 = 4 -390
exponential function definition
a function in which the variable occurs as part of the index or exponent; simplest have the form f(x) = a^x where a is positive and not equal to 1.
2^(-x) =
(1/2) ^ x [391]
When you put money in the bank, what do you call your balance? What do you call the money the bank pays you? What do you call the effect of this growth?
Your balance is your PRINCIPAL; what the bank pays you is INTEREST; how the money grows is COMPOUNDED since the interest is then itself added to the principal which then earns you more interest -395
What is the formula for the future value of an amount initially invested (present value) compounded at an interest rate per year of i for n years?
Future Value = Present Value x (1 + i) ^ n [396]
If you invest $5000 at 8% for 2 years, how much money will you have?
FV = PV x (1+i)^n
FV = 5000 x (1 + .08) ^ 2 = 5000 x (1.08)^2 = $5,832 -396
If I must save $10,000 for a boat in 4 years, and can get 8.5% per year return on my initial investment, how much should I invest today?
FV = PV x (1+i) ^n
10,000 = PV x (1.085)^4
Solve for PV => PV = $7,215.74
What is the formula for depreciation?
Future Value = Present Value x (1 - i) ^ n [399] This is the mirror image of the future value of an amount compounded, it's just that a depreciation rate is just a NEGATIVE interest rate.
If a^x = a^k, what do we know about x and k?
x = k [page 400]
What is a logarithm?
A logarithm in base 10 of any positive number is the power you would have to raise 10 to get that number. So the log of 100 is 2 since 10^2 is 100. The log of 1,000 is 3 and the log of 1,000,000 is 6 (for this special case, you can just count the zeros). -404
What is the log of 1?
0
What is the log of 100?
2
What is the log of 1000?
3
log(b)1
0
log(b)b
1
log(b)b^x
x
b^log(b)x
x
log1
0
log10
1
log10^x
x
10^logx
x
ln1
0
lne
1
lne^x
x
e^lnx
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