Algebra 2: Chapter 7

Question 1

what equation is used when modeling growth/decay situation?

what does everything stand for?

Answer 1

A(t) = a (1 +/- r)^t

A = final amount
a = initial amount
\+/- = growth (+), decay (-)
r = rate of increase (do percent in decimal form)
t = # of time periods

Question 2

what is the inverse of f(x) = x/8?

Answer 2

f^-1(x) = 8x

Question 3

when writing the inverse of a table of functions….

domain and range..?

Answer 3

write the domain and range in numerical order (least –> greatest), but graph it like you would the regular table of x and y values, except put the y-values as the x-values, and vice-versa

Question 4

what is the product property of logarithms?

when is this used?

Answer 4

log b MN = log b M + log b N (and vice-versa)

to express a SINGLE logarithm

Question 5

what is the quotient property of logarithms?

Answer 5

log b (M / N) = log b M - log b N

Question 6

what is the power property of logarithms?

Answer 6

log b A^p = P log b A

P = power

Question 7

what is the inverse property of logarithms?

Answer 7

log b B ^x = X or B^log b X = X

Question 8

how is a base of a logarithm function changed into a base of 10?

Answer 8

log b X = (log a X) / (log a B)

Question 9

how do you solve an exponential equation? (2 methods)

Answer 9

1. write them so all of the bases are the same
   ex. if B^x = B^y, then X = Y
2. take the log of BOTH sides
   ex. if A = B, then log A = log B

Question 10

what do you do when you have (X^a) ^b ?

Answer 10

you MULTIPLY the exponent, so it is… X^ab

Question 11

how do you solve logarithm equations?

Answer 11

if log b X = log b Y, then X = Y

Question 12

properties of e? (2)

Answer 12

• approx. 2.72

- it’s the base of the NATURAL log –> log e base = ln (not common log)

Question 13

what is the natural log?

Answer 13

ln —> log e base = ln

***Don’t write log e.

Question 14

which equation do you use for continuous compounds…?

what do the letters stand for?

Answer 14

A = Pe^rt

A = current amount
P = principal amount (initial amount)
r = rate (expressed in decimal form)
t = time (ie. amount of years, days, hours, etc.

Question 15

which equation is used to calculate half-lifes?

what does everything stand for?

Answer 15

N(t) = No e ^-kt

N(t) = amount REMAINING 
No = initial amount (grams)
-k = decay constant 
t = time

Question 16

when calculating half-lives, what do you do in order to find the decay constant (-k)?
[4 STEPS]

Use this for an example:

An element has a half-life of 15,000 years. How much of a 3gram sample is left after 12,000 years?

Answer 16

1. put everything equal to 1/2
   * 1/2 = 1e ^ -k(15,000)
2. give both sides ln
   * ln(1/2) = ln e (-k)(15,000)
3. ln e cancel each other out, so the equations is…
   * ln (1/2) = -15,000k
4. divide 15,000 by ln (1/2)
   * -k = 4.62 x 10^-5

Question 17

How to completely calculate half-lives (assuming you already found the decay constant)
[2 STEPS]
Use this for an example:

An element has a half-life of 15,000 years. How much of a 3gram sample is left after 12,000 years?

Answer 17

1. Plug everything in, into the given equation (N(t) = No e^-kt
   * N(t) = 3e^-4.62x10^-5 (12,000)
2. Evaluate.
   * N(t) = 1.72grams