Exponential & Logarithmic Properties
logₐ(1/a)
-1
logₐ(1/x)
-logₐx
ln1
0
log(b)1
0
log1
0
logₐ1
0
lne
1
log(b)b
1
log10
1
logₐa
1
Exponential Function
A function in which the exponent is a variable
one-to-one function
A function that matches each output with one input
inverse function
A function that reverses the effect of another function
natural logarithm
A logarithm with base 'e'
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)
Napierian logarithm
AKA natural logarithm, named after John Napier, a Scottish theologian and mathematician who discovered logarithms
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)
e
An irrational number called the natural base e is about 2.7183
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
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
exponential function
Includes a constant raised to a variable power, f(x) = b^x. The base b must be positive but cannot equal 1
compounded annually
Interest that builds on itself at 12 month intervals
Continuously growing populations
N = N0e^kt
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)=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)+c
Shifts the graph c units upward
change-of-base formula
State log16 32 as an expression using 2 base logarithms
irrational constant
The number 'e'. A number that repeats without pattern
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
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
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"
Exponential Decay
When a graph or function changes by decreasing amounts
Exponential Growth
When a graph or function changes by increasing amounts
half-life
a fixed period of time in which something repeatedly decreases by half
One-to-one Function
a function in which no second element repeats
natural base exponential function
a function of form f(x) = ae^rx
Exponential Function
a function of the form f(x)=ab×, where the coefficient a≠0, the base b>0 and b≠1
horizontal asymptote
a horizontal line that the curve approaches but never reaches
Asymptote
a line that a graph approaches but never touches
sound intensity
a measure of how much power sound transmits
Function
a set of ordered pairs in which no first element repeats
continuous
a smooth curve; there are no gaps in the curve for the domain
General Form of a Natural Log
aln (x-h) +k
Logarithm
b ^x=y log y=x b
y=log(base)bX is by definition equal to
b^y=X where b>0 and b≠1
Natural Log
e^x=y lny=x
The exponential function with base b
f(x)=b^x
The natural exponential function
f(x)=e^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)
continuously compounded interest
interest that builds on itself at every moment f(t) = Pe^rt
Between each level on a logarithmic scale
it increases by a power of 10 (2 levels = 100x)
Change of Base (natural logs): log(b)M
lnM/lnb
The common log logx means:
log(10)x
log(b)N-log(b)M
log(b)(M/N)
log(b)N+log(b)M
log(b)(MN)
plog(b)M
log(b)M^p
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
Express in logarithmic form: y=b^x
log(b)x=y
Common Logarithm
log(base)10X or log X. THEREFORE y=log X is by def'n 10^y=X
The natural log lnx means:
log(e)x
General Form
log-alog (x-h) +k b
Change of Base (common logs): log(b)M
logM/logb
common logarithm
logarithms with base 10
logₐm-logₐn
logₐ(m/n)
aⁿ=b
logₐb=n
logₐm+logₐn
logₐmn
logₓa
logₑa/logₑx
sound level
measured in units called decibels (dB); provides a scale that relates how humans perceive sound to a physical measure of its power
logₐaⁿ
n
logₐxⁿ
nlogₐx
The Power Rule: log(b)M^p
plog(b)M
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 = ?)
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 = ?)
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.
pH
take negative log of scientific notation
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
Hydrogen Ion Concentration
take pH and make it the negative exponent of 10, enter into calculator to find scientific notation
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
exponential growth
the graph of an exponential function with a base greater than 1
logarithmic function
the inverse of an exponential function
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)
Base
watever is being raised to a power
10^logx
x
b^log(b)x
x
e^lnx
x
lne^x
x
log(b)b^x
x
log10^x
x
General Form of Exponential
y=a times b^ x-h +k
Express in exponential form: log(b)x=y
y=b^x
