PreCalc Chapter 3 Test Flashcards
what are the logarithmic models?
y=a+bLnx
y=a+blogx
what is the equation for exponential growth?
decay?
what does each variable stand for
A=Ie^kt and A=le^-kt A is what you have (dependent) I is your initial amount e is 2.71828 (constant/base/multiplier) k is constant of exponentialism (constant coefficient/growth rate) and t is time (independent)
An exponential growth model has the form ______, and an exponential decay model has the form_________.
y=ae^bx or A=Ie^kt
y=ae^-bx or A=Ie^-kt
in probability and statistics, Gaussian models commonly represent populations that are ___________ _____________.
normally distributed
a logistic growth model has the form
y=a/1+be^-rx
a logarithm is an _____
exponent
logarithmic and exponential equations are _______
inverses
the parent log has an assumed base of ____
the natural log has an assumed base of __
10 (common log)
e
what does e=
2.71821
logarithmic graphs have a ____ asymptote (__=#) while exponential graphs of a ____ asymptote (__=#)
vertical x
horizontal y
is e a variable?
no, it is a constant
what is true of all logarithmic graphs?
how do you restrict the domain?
how do they look?
they all pass vertical line test, always have x-intercept and asymptote
you cannot take the log of 0 or a negative number
boomerang
how you know if an equation is exponential
it has a variable as an exponent
what would you do with
y=2^-x^2
take the square of the number first then multiply by the implied negative one
how would you simplify 3^x-2
3^x3^-2
3^x(1/9)
like terms have the same ____ and same _________
base, exponent
polynomial functions are examples of ___ functions
algebraic
exponential and logarithmic functions are examples of nonalgebraic functions, also called _____ functions
transcendental
you can use the _______ Property to solve simple exponential equations
one-to-one
the exponential function f(x)=e^x is called the ______ _____ function, and the base e is called the ______ base
natural exponential, natural
to find the amount A in an account after t years with principal P and an annual interest rate r compounded n times per year, you can use the formula______________.
A=P(1+r/n)^nt
to find the amount A in an account after t years with principal P and an annual interest rate r compounded continuously, you can use the formula _____________.
A=Pe^rt
the inverse function of the exponential function f(x)=a^x is called the ______ function with base a
logarithmic
the common logarithmic function has base
10
the logarithmic function f(x)+lnx is called the ______ logarithmic function and has base _________
natural, e
the inverse properties of logarithms state that logaA^x=x and
a^logaX=X
to one-to-one property of natural logarithms states that if Inx=Iny, then_______________
x=y
the domain of the natural logarithmic function is the set of __________ ________ __________.
positive real numbers
all real numbers such that x is greater than 0
to evaluate a logarithm to any base, use the ______ formula
change-of-base
the change-of-base formula for base a is given by logaX=__________
logbX/logbA
you can consider logaX to be a constant multiple of logbX; the constant multiplier is ______________
1/logbA
what is the product property of logs
goes to addition
log4X*y^2=log4X+log4Y^2
what is the quotient property of logs
to difference
log7 x^3/y=log7X^3-log7Y
what is the power property of logs
logx^4=4*logx
(make sure you only bring it out if it is for the WHOLE THING
while doing the change-of-base formula, do you use the natural log or the common log
it does not matter as long as you are consistent
how can you make powers roots and roots powers
the cubed root equals ^1/3 and so on
how do you solve exponential or logarithmic (or any, really) equation
use inverse operations
your equation is done when _ is by itself
x
what is the log of 625 to base 5
what is the log of .001
4, -4
when solving an equation, it is important to check for _____________ by plugging your solutions back into the original equation
extraneous
when you are told to find an exponential model, make sure you have all variables expect____
the x and y, or the A and t
for like terms with the same base and exponent, when multiplying, add exponents but ______________________ the bases
do not do anything with the bases
asymptotes begin with _________________________.
x= or y=
if you have a log that can be taken by reducing the number, what do you do
reduce it, multiply it by the number you reduced it by, separate by addition, and finish
remember to use ________ when you are solving logs to indicate separation
parentheses
when simplifying logs, make sure to use parentheses between subtraction and addition and also make sure that the log of a certain number–make sure that number cannot be divided by anything to get a whole number answer. if it can, multiply and add based on rules of expansion
..
when solving an exponential equation, take the ___ of both sides
log
half life equations are decay, meaning k is
negative
