algebra 2b - unit 2: out of the woods lesson 6-10
what is the value of log2 8?
3
all of the following expressions are equivalent to 5⋅15t5⋅15t. However, only one expression has been rewritten by using exponent property #4, (b/c)^x = b^x/c^x. which one is it?
5*30^t/2^t
in order to write log5 25 = 2 in exponential form, it is necessary to convert it to the format b^x =y. what are the values of b, x, and y for the above equation?
5, 2, 25
in order to write 6^3 = 216 in logarithmic form, it is necessary to convert it to the format logb x = y. what are the values of b, x, and y for the above equation?
6, 216, 3
all of the following expressions are equivalent to 6 ⋅ 15^t. however, only one has been rewritten using exponent property #3
6^t*15^t/6^t-1
consider the function f(x) = 3x − 16. what is x equal to?
7
which expression(s) is (are) equivalent to 7 ⋅ 54^t? select all that apply.
7 * 6^t * 9^t 7 * 2^t * 3^3t 7 * 6^t *3^2t 7 * (2 * 3^3)^t
one method to find the inverse function of y = 3^x is to switch x and y, and then solve for y. in this case, the first step in that process results in x = 3^y. what should you do next to write the inverse function?
rewrite x = 3^y in logarithmic form to get log3 x = y.
consider the function f(x) = 4x^2 − 7 defined for the domain 0 ≤ x < 20. which answer shows the correct steps to find f^-1?
step 7: f^1(x) = √x+7/2
consider the function f(x) = −1/2(x^3 − 6) + 9 on the domain (−∞,∞). which answer shows the correct steps to find f^-1(x)?
step 8 : f^-1(x) = 3sqrt(24-2x)
examine the equation e^x = sqrt(x-3)(y+4)(z-5). which option correctly isolates y?
y = e^2x/(x-3)(z-5) - 4
which equation has the same meaning as ln x = 5?
e^5 = x
which equation has the same meaning as ln 5 = x?
e^x = 5
given f(x) = 10x + 1, what is f^−1(x)?
f^-1(x) = log(x - 1)
given f(x) = 2^x, what is f^−1(x)?
f^-1(x) = log2 x
if f(x) = 3 * 5^x, what is f^-1(x)?
f^-1(x) = log5 (x/3)
lesson 8
inverse functions
if e^−3 ≈ 0.0498, which equation is also true?
ln (0.0498) = -3
what does the notation f^-1 mean?
f^-1 is the inverse of the function f.
consider the function f(x) = 3√x-3 with a domain of 3 ≤ x < ∞. what is the range of f^-1(x)?
3 ≤ f^−1(x) < ∞
use the properties of exponents to rewrite the expression (x^2*x^5/x^-1*y^4 in the form x^a * y^b. what are the values of a and b?
3, 1
in order to write 3^5 = 243 in logarithmic form, it is necessary to convert it to the format logb x = y. what are the values of b, x, and y for the above equation?
3, 243, 5
in order to write log3 2187 = 7 in exponential form, it is necessary to convert it to the format b^x =y. what are the values of b, x, and y for the above equation?
3, 7, 2187
suppose the invertible function φ(x) has the domain (−5, 11) and the range (−12, 1). what is the range of φ^−1(x)?
(-5, 11)
what is the value of log 0.01?
-2
consider the function f(x) = tanx with a domain of 0 < x < π/2, where x is in radians. what is the range of f^-1(x)?
0 < f^-1 (x) < π/2
consider the function f(x) = cos x + 1 with the domain −π ≤ x ≤ 0. a: what is f^1? b: what is the domain of f^1?
0.25 of 1 a: f^-1(x) = arccos(x - 1) b: −2 ≤ x ≤ 0
use the following graph of the function f(x) = 2x^3 − 4 to fill in the blanks.
0.25 of 1 blank 1: horizontal line test blank 2: is not blank 3: there is a horizontal line that intersects blank 4: more than one place
which functions are one-to-one on the domain (−∞,∞)? select all that apply.
0.25 of 1 f(x) = x^2 + 1 g(x) = x^3 - 5
use the half-life function q(t) = 3500..... what is the half-life, h..
0.32
use the exponential expression 4/3*9^t/2+1 to answer the question. are the following expressions equivalent to the original?
0.5 of 1 a: yes, this expression is equivalent. b: no, this expression is not equivalent. c: no, this expression is not equivalent.
natalia has 40 grams of a radioactive substance. the amount of substance left after n cycles of its half-life is given by the formula a = 40(2)^−n. if natalia wants to rearrange the formula to write nn in terms of aa, which steps should she take? match each step with the action she should take in that step.
0.5 of 1 step 1: divide both sides of the equation by 40 step 2: divide both sides of the equation by 20 step 3: rewrite the equation in logarithmic form. step 4: simplify each side of the equation, if possible.
lorelei is studying a population of rabbits. after accounting for predators, she finds that the population doubles in size every 2 months. the initial population she studied had a total of 4 rabbits. after t months, the population, p, can be expressed as p = 4(2)^t
1/4p
which of the following expressions is/are equivalent to 18 * 6^-2x+1? select all that apply.
108/6^x 3 * 36^1-x 3 * 36/36^x
which equation has the same meaning as log x = 7?
10^7 = x
consider the compound interest equation b(t) = 100(1.1664)^t. what is the interest rate, r, written as a percentage?
16
velocity is defined as change in position over change in time, and it is given by the formula v = s2-s1/t2-t1? what is s2....?
165
consider the function f(x) = x^3+1/2, and its inverse, f^-1 (x) = 3√2x-1. what is the correct solution?
2
what number, rounded to the nearest hundred thousandth (five decimal places), is represented by e?
2.71828
all of the following expressions are equivalent to 27^t−1 ⋅ 2^t. however, only one has been rewritten using exponent property #4?
27^t-1, 16^t/8^t
which option is equivalent to 2^x * 3^7+y = 2^5 * 3^y?
2^x-5 * 3^7 = 1
which of the following expressions is/are equivalent to 3 ⋅ 36^x? select all that apply.
3 * 6^2x 2^2x * 3^2x+1 12^x * 3^x+1
use the compound interest equation b(t) = 124... how many times per year is the interest calculated?
4
in the half-life function q(t) = 28550 * ((3/4)^h)^(t/h)... use this tool for calculating h to find your answer
XXX 2.40 idk sorry
match the operation to its inverse operation.
add : subtract subtract : add multiply : divide divide : multiply raise to the power n : raise to the power 1/n take sin x : take arcsin x
which equation has the same meaning as logb a = c?
b^c = a
which properties of exponents have been used to rewrite the expression 6^2t as (12^t/2^t)^2? select all that apply.
b^x/c^x = (b/c)^x (b^x)^y = b^xy
use the following graph of the function f(x) = −3x^2 + 2x + 6 to fill in the blanks.
blank 1 : horizontal line test blank 2 : is not blank 3 : there is a horizontal line that interest blank 4 : more than one place
consider the function f(x) = 3 cos(x-pi/4)...
XXX a: −∞ < x < ∞ b: the domain of f(x) must be restricted so that it matches the domain of f^−1(x). b: both functions f(x) and f^−1(x) must be one-to-one, and f^−1(x) is not one-to-one unless the domain is restricted.
a bug spray's effectiveness decreases over time. after h weeks, it retains only half of its effectiveness, and this trend continues. a function that represents this is .... a: what is h to the nearest hundredth? b: which option correctly shows the function rewritten in the form..
a : h = 0.39 b: q(t) = 233,280 * (1/2)^t/0.39
the area of a particular parallelogram is given by the formula a = x^2 - bx + c. how can you rearrange this formula to correctly find the x-value that will yield the maximum area?
a = (x - b/2)^2 + 4c - b^2 / 4
tadashi is a landscape architect, and he is designing a pool for a client's backyard. the pool will be twice as long as it is wide, and it will have a 3-foot walkway surrounding it on all sides. which equation represents the total area that the pool surface and walkway will take up in the backyard?
a(w) = (w + 6)(2w + 6)
use the exponential expression 9/4^8 * (256)^t/8+2 to answer the questions. a: which option correctly shows the expression rewritten in the form a * b^t? b: if this equation represents population growth of a type of bacteria, where t is the number of days, how quickly does the bacteria grow?
a: ...step 6: 9 * (2)^t b: the number of bacteria doubles every day.
use the following formula for the volume of a regular tetrahedron to answer the questions. a: which option correctly shows how the given formula can be rearranged to isolate the radius, r? b: what is the value of the radius when the volume is 9 cm^2?
a: 3√6v * 6√2 = r b: 3√2
consider the function f(x) = √x-5 + 5 for 5 ≤ x < ∞.
a: 5 ≤ x < ∞ b: the domain of f^−1(x) must be restricted so that it matches the range of f(x). b: both functions f(x) and f^−1(x) must be one-to-one, and f^−1(x) is not one-to-one unless the domain is restricted.
use the compound interest equation b(t) = 500(1.001)^12t to answer the questions. a: which option correctly shows the equation rewritten in the form b: what is the interest rate, r, written as a percentage?
a: b(t) = 500(1 + .012/12)^12t b: 1.2%
the formula for the value of a car, c, after t years of depreciation of its initial value, v, at a rate of r is c = v(1 - r)^t. a: how can you rewrite the formula to correctly isolate v? b: what was the initial value of the car (to the nearest dollar) if it depreciated 10% each year for 10 conservative years and has a current value of $8,019.60?
a: c/(1-r)^t = v b: $23,000
use the equation f(x) = 4x+5/6 to answer the questions.
a: f^-1 (x) = 6x-5/4 b: find f^-1(8) c: 10.75
use the invertible function f(x) = √x+3 − 2 to answer the questions. a: what is f^-1(x)? b: what is the domain of f^-1?
a: f^-1(x) = x^2 + 4x + 1 b: : −2 ≤ x < ∞
the function f(x) = 3√(x^2 + 1) − 4 is invertible on the domain [0,∞). a: what is f^-1(x)? b: what is the domain of f^-1(x)?
a: f^−1(x) = √(x+4)^3-1 b: [−3, ∞)a
dr. valenzuela wants to find the half-life of a radioactive substance. she tried to rewrite the function. a: what is h to the nearest hundredth? b: which option correctly shows the function rewritten in the form
a: h = 0.43 b: q(t) = 9375(1/2)^t/0.43
dr. norton wants to find the half-life of a medication for his patients. he tried to rewrite the function. a: what is the value of h to the nearest hundredth? b: which option correctly shows dr. norton's function in the form
a: h = 0.50 b: q(t) = 25600 * (1/2)^t/0.50
the difference in volumes between a cube-shaped box and a sphere with a diameter equal to the side length of the box is given by the formula v = s^3 - 1/6pis^3. a: which option correctly isolated s? b: what is the side length of the box if the difference in volumes is 12 cubic inches?
a: s = 3√v/1-1/6pi b: 2.93 inches
use the exponential function f(t) = 1.02^t to answer the questions. part a: what is the equation of the horizontal asymptote in this function? part b: does this function represent exponential growth or decay? part c: what is the initial value of the function, f(0)?
a: y = 0 b: exponential growth (increasing) c: f(0) = 1
in a particular science experiment, the number of bacteria is represented by the function p(t) = 100e^t + 3, where t is time in days. a: what is the equation of the horizontal asymptote? b: what number represents the initial number of bacteria, p(0)? c: what value of t should be used if you check the population after 60 hours?
a: y = 3 b: 103 c: 2.5
the equation p(t) = 12e^t + 7, where t is time in months, represents a continuously growing population of a frog species. a: what is the equation of the horizontal asymptote? b: what number represents the initial number of frogs, p(0)? c: what value of t should be used when the population is checked after 2.5 years?
a: y = 7 b: 19 c: 30
consider the function f(x) = -√x+2 + 5 for −2 ≤ x < ∞.
a: −∞ < x ≤ 5 b: the domain of f^−1(x) must be restricted so that it matches the range of f(x). b: both functions f(x) and f^−1(x) must be one-to-one, and f^−1(x) is not one-to-one unless the domain is restricted.
lesson 9
logarithms, part 1
lesson 10
logarithms, part 2
match each equation in logarithmic form to its equivalent equation in exponential form.
logx e = 3 : x^3 = e log3 e = x : 3^x = e ln x = 3 : e^3 = x ln 3 = x : e^x = 3
examine the system of equations {x+2y+z=3.... which observation about the system of equations is accurate?
the system is potentially solvable because there is an equal number of equations and unknowns
is f(x) = 4x^2 − 3 a one-to-one function on the domain (−∞,∞)?
no, is not a one-to-one function on (−∞,∞) because there is an output value that has more than one corresponding input value.
minh is building a wooden box to hold all of her dinosaur figurines. she wants the base of the box to be in the shape of a square, and she has a total of 1 square meter of wood to use when making the box. what dimensions of the box will maximize the volume? this problem is an example of a(n) ____ problem.
optimization
the formula that gives the total amount of money, a, when an amount of money, p, is invested at an annual rate r for t years with interest compounded n times per year is a = p(1 + r/n)^nt. which option correctly shows the given formula rearranged to find p?
p = a / (1 + r/n)^nt
le fromage is an artisanal cheese shop in paris, arkansas. they sell manchego for $2.12 per ounce, and parmigiano reggiano for $1.77 per ounce. which equation for r correctly describes the revenue le fromage makes if it sells x ounces of manchego and y ounces of parmigiano reggiano?
r = 2.12x + 1.77y
match each equation in exponential form to its equivalent equation in logarithmic form.
r^3 = 7 : logr 7 = 3 7^3 = r : log7 r = 3 r^7 = 3 : logr 3 = 7 3^r = 7 : log3 7 = r
lesson 6
rearranging formulas and isolating variables
lesson 7
rewriting exponentials
which method should you use in order to determine the correct value of log5 25?
solve for 5^x = 25
a bank account balance, b, for an account starting with $100 dollars, earning an annual interest rate of 0.2%, and otherwise having no deposits or withdrawals for tt years, can be calculated as b = 100(1.002)^t
step 1: divide both sides of the equation by 100. step 2: take the logarithm with base 1.002 of both sides of the equation. step 3: simplify each side of the equation, if possible.
use the following formula which represents the total resistance in an electrical circuit with two resistors placed in parallel, to answer the question. which option correctly rearranges the formula to isolate r1?
step 1: r(tot) = r1r2/r1+r2 step 6: r(tot)r2 / r2-r(tot) = r1
the formula for the value of a car, c, after t years of depreciation of its initial value, v, at a rate of r is c = v(1 - r)^t. how can you rearrange the given formula to correctly isolate the depreciation rate, r?
step 4: r = 1 - tsqrt(c/v)
use the following formula for slope to answer the question. which option correctly rearranges the formula to isolate x2?
step 4: x2 = x1 + y2-y1/m
use the equation y = x^4 - 2x^2b^2 + b^4 to answer the question. how can you rewrite the given formula to correctly isolate x?
step 5: +-sqrtb^2 +- sqrt(y) = x
use the following formula, which gives the area of a triangle, and where s is the semiperimeter (half of the perimeter of the triangle), to answer the question.
step 5: c = s - a^2 / s(s-a)(s-b)
use the following vertical motion model, where h is the height in feet of a launched object t seconds after it is launched at an initial velocity, v, from an initial height of s feet. which option correctly shows how the given formula can be rewritten in vertex form to highlight the maximum height of the object that is launched?
step 5: h = -16(t - v/32)^2 + v^2 + 64s/64
the formula that gives the total amount of money, a, when an amount of money, p, is invested at an annual rate r for t years with interest compounded n times per year is . which option correctly shows how the given formula can be rearranged to find r?
step 5: n(nt√a/p-1) = r
the distance, d, between two points, (x1, y1) and (x2, y2) can be found using the formula... how can you rearrange the given formula to correctly find y2?
step 5: y1 +- sqrt(d^2 - (x2 - x1)^2) = y2
which option shows how to correctly isolate the variable x in the equation bd/3y = 5b/6x-d?
step 6: x = 15y+d^2 / 6d
which option correctly shows how qpx^2 = 16xd^4 - 3npx can be rearranged to isolate d?
step 7: +4√p......
use the invertible function f(x) = 5√7-x + 3 to answer the question. which option shows the correct steps to find f^-1?
step 7: f^-1(x) = 7 - (x - 3)^5
the quadratic formula states that .... how can you rearrange this formula to correctly find b?
step 8: b = -(ax^2 + c ) / x
which answer choice gives the solution set of the system of inequalities -5 < x and y >= 2?
the set of all ordered pairs whose x-coordinate is in the interval (-5, infinity) and whose y-coordinate is in [2, infinity).
one exponential formula for bacterial growth could be.... how can you rearrange the given formula to correctly find x0?
x0 = x/e^kt
which expressions are equivalent to x^7? select all that apply.
x^3 * x^4 (x^21)^1/3 x^15/x^8
a quadratic equation has a vertex at (-3/2, 7/2) and a y-intercept of 8. which equation(s) could describe this quadratic equation?
y = 2x^2 + 6x + 8 y = 2((x + 3/2)^2 + 7/4)
when a certain quadratic equation is graphed, it opens upward and has zeros at x = 3 and x = -2. which of the following equations could this graph represent?
y = x^2 - x - 6 y = (x - 3)(x + 2) y = (x - 1/2)^2 - 25/4 y = 3x^2 - 3x - 18
use the function f(x) = sin x to answer the question. which option represents a restricted domain that allows the function f to be invertible?
−π/2 ≤ x ≤ π/2 0 < x < π/2 −π/2 < x < π/2
use the function f(x) = 4x^2 − 3x + 1 to answer the question. which option(s) represent a restricted domain that makes f invertible?
−∞ < x < 0 −∞ < x ≤ −3/8 3/8 < x < ∞ 3/8 ≤ x < ∞ −3/8 < x < 3/8
consider the function f(x) = −x^2 + 6 on the domain [0,∞). use the range of f(x) to determine the domain of f^−1(x). what is the domain of f^−1(x)?
−∞ < x ≤ 6