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Level 581

Exponential & Logarithmic Functions


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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 =