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domain-all real numbers
Exponential function is
a function in the form y = b^x where b the base is a constant positive value but not equal to 1 and x is a variable exponent
To evaluate and exponential function
is the same as other functions, substitute the given value for the variable and simplify.
An exponential function can be graphed by
making a table to find and plot points on the graph. Choose several values for x and keep them simple
The graphs of exponential functions like y = a^x where a > 1 all have the same basic shape
The graph grows rapidly for positive values of x, and shrinks gradually approaching zero for negative values of x. The x-axis in this case is a horizontal asymptote.
The larger the base of an exponential function is
the more rapidly it grows for positive values of x, and the more rapidly it approaches zero for negative values of x
if the base of an exponential function is between 0 and 1
you can use the properties of exponents and rewrite the power with it reciprocal with a negative exponent.
the base of an exponential function can never
be a negative number it is always positive and not equal to 1
To solve an exponential equation
use the properties of exponents to write each side of the equation with like bases or common bases and then set the expression in the exponents equal to each other and solve for the variable
what is an exponential equation
an equation with variable in the exponents such as the case with exponential functions
Negative power property:
a / b^-x = b / a ^ x
that means the exponents of the expression are also equal
if two expressions are set equal to each other and they have the same bases
compound interest formula
A = p (1+ r/n)^nt
the number e
also known as Napier's number, is the base of a natural logarithm, approximately equal to 2.71828
the number e can be approximated with the expression
(1 + 1/n)^n where the larger n is the closer the expression gets to the value of e
The number e is connected to
the slope of tangents to the exponential curve and has applications in population growth and decay and many others
Logarithms and exponential functions
are inverses of each other. Logs can be used to untangle expressions involving exponential functions.
is an exponent;
a log or logarithm
if y is the logarithm with base b evaluated at x then
b is the base raised to some exponent (y) that evaluates to x
a log written without a b
is the common log, y = log x where the base is understood to be 10
never take the log
of a negative number
whenever converting from logarithmic form to exponential form remember
a log is an exponent meaning the number the log represents is the exponent of the base
the inverse of the exponential function
is always equal to the power z is raised to
a logarithm with a base z evaluated at z raised to some power
the x - intercept and the approximately how high it gets.
Since all logarithms have the same basic shape to graph them all you need to really know is
To find the x-intercept of a logarithm
set the logarithm equal to zero and solve for x
the properties of exponenets
properties of logarithms are analogous to
The log of a product
is the sum of the logs
The log of a quotient
is the difference of the logs
is equal to 1
log with base b evaluated at b
is equal to 0
log with base b evaluated at 1
is y * log with base b evaluated at x
log with base b evaluated at (x raised to the y power)
is not the same at the sum of logs
the log of a sum ; log (3a + b)
the quotient of two logarithms (log a / log b)
is not the same as the log of a quotient
the natural log
has e at its base and is denoted (ln x) meaning the log with base e, evaluated at x
to combine logarithms into a single logarithm
use the properties of logarithms in reverse
to simplify complicated logs,
use the properties of logarithms
What are the buttons for evaluating logarithms on a calculator
[log] for log base 10, and [ln] for the natural log with base e
the change base theorem
is a theorem that provides a formula to easily change any logarithms base to another base making them easier to manipulate
what is the change base formula
log with base b of x is equal to the log with base a of x divided by the log with base a of b
is the natural log (base e) evaluated at A
the Richter scale
is a logarithmic scale found by taking the log of the intensity of the earthquake divided by the intensity of the normal seismic activity.
the distance modulus formula
is a formula for calculating large distances, m = 5 log r-5, where m is the magnitude of a starts brightness, and r is the stars distance measured in parsecs a value approximately 3.3 light years
to solve equations involving logarithms
convert the logs in one big log and convert them to exponential form and solve for the variable. When an equation has logs of logs you must do the untangling process more than once.
if the logs are equal
than the stuff the logs are evaluated at are equal too.
The exponential function f(x) w/ base a is written as...(full formula)
having the same value
There are _________ extraneous answers, only logarithmic functions
What does e equal
(Steeper Exponential Growth)
What is the application (formula) to use when the interest is compounded monthly, daily or yearly?
What is the application (formula) to use when the interest is compounded continuosly?
What does A stand for?
Amount of $ in an account (BALANCE)
What does P stand for?
amount of $ you invest to start (Principle)
interest rate (in decimal form)
What does R stand for?
time in years
What does T stand for?
What does n stand for?
# of times interest is calculated/ year (compounding)
If "a" is negative it means...
the ID point is an up/down flip
If "x" is negative...
y-axis flip (changes shape only)
x (the log(5) cancels)
a log ^a(x)=
When finding the ID point, you start counting from
log (subscript a)b= log(b)/log(a)
What is the change of base formula?
What is the product property?
What is the quotient rule?
What is the power property?