Logarithmic Functions
In mathematics, the logarithmic function is an inverse function to exponentiation
Domain and range of logarithmic functions
We know that is defined only when >0. So the domain is the set of all positive real numbers. We can see that can be either a positive or negative real number (or) it can be zero as well. Thus, can take the value of any real number.
Note:
The domain of function = is >0 (or) (0,∞).
The range of any log function is the set of all real numbers ().
Logarithmic graph
We know that exponentials and functions are inversely proportional to each other, and so their graphs are symmetric concerning the line =. Also, note that =0 when =0 as =(1)=0 for any . Thus, all such functions have an -intercept of (1,0). A logarithmic function doesn’t have a -intercept as 0 is not defined.
Properties of logarithmic graph
>0 and ≠1.
The logarithmic graph increases when >1 and decreases when 0<<1.
The domain is obtained by setting the argument of the function greater than 0.
The range is the set of all real numbers.
Graphing logarithmic functions
Before plotting the log function, just have an idea of whether you get an increasing curve or decreasing curve as the answer. If the >1 then the curve is increasing, and if 0<<1, then the curve is decreasing.
Here are the steps for graphing logarithmic functions:
Find the domain and range.
Find the vertical asymptote by setting the argument equal to 0. Note that a function doesn’t have any horizontal asymptote.
Substitute some value that makes the argument equal to 1 and use the property (1)=0. This gives us the -intercept.
Substitute some value that makes the argument equal to the base and use the property ()=1 This would give us a point on the graph.
Connect the two points (from the last two steps) and extend the curve on both sides relative to the vertical asymptote.
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Which rule describes the relationship between the ex and Y coordinates on the following graph
choose 1 answer:
A) y = x
B) y = 2x -2
(PICTURE INCLUDED WHOEVER GETS IT RIGHT FIRST GETS MARKED BRAINLIEST)
Answer:
A, y = x
Step-by-step explanation:
Slope is equal to one
Y-intercept is origin
Answer:
A.) Y=X
Step-by-step explanation:
It intercepts with origin
Jamie's sister loaned her $15. She now has at least $50 to purchase new boots. What is m, the amount of money Jamie had before her sister loaned her the money? Write an inequality to represent this situation.
15m ≥ 50
15m > 50
m + 15 ≥ 50
m + 15 > 50
Answer:
m + 15 ≥ 50
Step-by-step explanation:
Jamie's sister gave her 15 dollars on top of what she already had. This means that it was added to her own money, m. Also, the statement says that she has at least 50; this means that she has an amount equal to or greater than 50. Therefore, the third option, m + 15 ≥ 50, is correct.
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Gracie has made 2 gallons o punch that contains 50% juice. After Gracie pours out some of her mixture and replaced with an equal amount of pure 100% juice, she has 2 gallons of punch that contains 65% juice. How many gallons of the original mixture did Gracie pour out? Express your answer as a common fraction.
Gracie poured out 3/10 gallons of the original mixture.
What is percentage?In mathematics →
a percentage is a number or ratio expressed as a fraction of 100.it is often denoted using the percent sign "%".Given is that Gracie has made 2 gallons o punch that contains 50% juice. After Gracie pours out some of her mixture and replaced with an equal amount of pure 100% juice, she has 2 gallons of punch that contains 65% juice.
Assume that she pour out {x} gallons of original mixture. We can write -
{50% of 2} - {x} = {(100 - 65)% of 2}
{50/100 x 2} - {x} = {35/100 x 2}
{1/2 x 2} - {x} = {35/100 x 2}
1 - x = 7/10
1 - 0.7 = x
x = 0.3
x = 3/10
Therefore, Gracie poured out 3/10 gallons of the original mixture.
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can someone pls help
Answer:
10 walnuts..
Step-by-step explanation:
12 - 2 = 10
Answer:
10?
Step-by-step explanation:
12-2=10
Dean Wormer wants to compare the grade point average of athletes at Faber College with the overall grade point average at Faber College. She randomly selects three students from each team and determines that their combined grade point average, 2. 3, is the same as that of all students at Faber. In this example, what is the population
yes it is the same as that of all students at Faber GPAs as 2.32 and lower will mean that a track team athlete will be recommended for extra tutoring and population is 2.3* student number
What is the definition of a population?
In ordinary usage, a population is a distinct group of individuals with shared citizenship, identity, or characteristics. In statistics, a population is a representative sample of a larger group of people (or even things) with one or more characteristics in common.
When the distribution is normal, we use the z-score formula.
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
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Clara and erin volunteered at an animal shelter a of 9 1/5 hours. Clara worked for 41/3 hours. How many hours did erin work?Yon can use fraction strips to solve this problem
If $10,000 is invested at x percent simple annual interest for n years, the total amount of interest would be F = 10,000n(x/100)
What is simple interest?Simple interest is calculated based on the principle of a loan or the initial deposit into a savings account. Simple interest doesn't compound, so a creditor will only charge interest on the principal sum, and a borrower will never be required to pay interest on interest that has already accrued.
F = P(1+rt), formula for simple rate of interest
Here,
=> Future value is F
Present value is P
=> Rate is R
=> Time is t
Substituting x for r
And, n for t
Also, 10,000 in for P
F = 10,000(1+xn)
The formula tells us principal + interest,
F = 10,000(xn)
Now, to get the x into terms of a percentage,
F = 10,000(n * x/100)
F = 10,000n(x/100)
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Joanne is campaigning for class president and plans to distribute some campaign materials. 20 flyers and 16 buttons. She wants each classroom to receive an identical set of campaign materials without having any materials left over. What is the greatest number of classrooms Joanne can distribute materials to?
The greatest number of classrooms is 4, Joanne can distribute materials
5 flyers and 4 buttons.
What is the greatest common factor?The greatest number that can divide the numbers in equal parts.
It is called as greatest common factor or GCF.
Given:
Joanne is campaigning for class president and plans to distribute some campaign materials.
20 flyers and 16 buttons.
She wants each classroom to receive an identical set of campaign materials, without having any materials left over.
In order to find the greatest number of classrooms, Joanne can distribute materials:
We have,
the number of flyers = 20.
Number of buttons = 16
We need to find the GCF.
GCF of 16 - 1,2,4,8,16
GCF of 20-1,2,4,5,10,20
Clearly, 4 is the GCF.
Therefore, total number of classrooms = 4
Number of flyers distributed to each classroom
= Total number of flyers / Total number of classroom
= 20/4
= 5
Number of buttons distributed to each classroom
= Total number of buttons/ Total number of classrooms
= 16/4
= 4
Therefore, the Joanne can distribute 5 flyers and 4 buttons to each classroom.
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PLEASE HELP!!
The function f(x) has a vertical asymptote at x = [BLANK].
Answer:
-4
Step-by-step explanation:
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Answer:
B: weather occurs because the climate is constantly changing
Step-by-step explanation:
Given cos 0= -1/2 and angle 0 is in quadrant two what is the exact value of sin 0  in simplest form
Answer:
[tex]-\sqrt{7/4}[/tex]
Step-by-step explanation:
[tex] \sin( \theta) = \frac{ \sqrt{3} }{2} [/tex]
The number y of duckweed fronds in a pond after t days is y=a(1230. 25)t/16, where a is the initial number of fronds. By what percent does the duckweed increase each day? Round your answer to the nearest whole number
The duckweed increases by a percentage each day which is calculated by the equation
(1230.25t/16)/a - 1.
This equation can be further simplified to 1230.25t/16a - 1 and then multiplied by 100 to get a percentage.
So the percentage increase of duckweed each day is calculated by multiplying 1230.25t/16a - 1 by 100.
1230.25t/16a×100
This will give us the percentage of increase each day, rounded to the nearest whole number.
For example, if the initial number of fronds is 1000 and the number of days is 2, the percentage increase of duckweed will be (1230.25(2)/16(1000)) - 1 = 24.6%, rounded to the nearest whole number which is 25%.
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Use the graph of the function to find the domain and range of f (Enter your answers using interval notation)
In the given graph the domain and range of function y=f(x) is
Domain: (-2,2]
Range : [-4, 5)
What is domain and range?A function's domain is the set of all possible inputs to the function. This means it can be any number that can be entered into a function. For the majority of functions, this will be any number that can be substituted for the letter x. Except for a few exceptions such as square root functions and rational numbers, your domain will almost always be all real numbers.
The set of all possible outputs of a function is referred to as its range. This is usually represented by the letters y or f(x). The range is any number obtained by substituting any number for x.
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A triangle has a height that is 4 inches shorter than its base. If its area is 30 square inches, find the base and height.
Answer:
The height of the triangle = 6 inches
The base of the triangle = 10 inches
Step-by-step explanation:
The formula for the Area of a Triangle = 1/2 × Base × Height
Area = 30 in²
h = b - 4
Hence
30 = 1/2 × b × b - 4
Cross Multiply
30 × 2 = b² - 4b
60. = b² - 4b
b² - 4b - 60
We factorise the Algebraic expression to find b
b² - 4b - 60 = 0
b² - 10b + 6b - 60 = 0
b(b - 10) + 6(b - 10) = 0
(b - 10) + (b + 6 ) = 0
b - 10 = 0 , b = 10
b + 6 = 0, b = -6
We are going to take the positive answer of b as the value of b
b = 10 inches
Solving for h
h = b - 4
h = 10 - 4
h = 6 inches
Therefore, the height of the triangle = 6 inches
base of the triangle = 10 inches
What is the difference between a difference of two squares and a perfect square trinomial?
Depending on whether a and b are real numbers, they can factor as (a + b) (a + b) or (a - b)(a - b). Forms like (a + b)(a - b) are unique products also known as the difference of squares.
what is perfect square ?When you square an integer, you get perfect squares. The number 81, for example, is a perfect square since it can be derived by squaring 9: 99=81. Because 12 can be squared to make 144, 144 is a perfect square. 13 may be squared to make 169, which is a perfect square. Mathematics-related language. Informally: When an integer (a "whole") number is multiplied by itself, whether the result is positive, negative, or zero, the result is referred to as a square number, a perfect square, or simply "a square." All square numbers are therefore 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and so on.
given
Perfect square trinomials, also referred to as trinomial squares, are the squares of binomial expressions.
Depending on whether a and b are real numbers, they can factor as (a + b)(a + b) or (a - b)(a - b). Forms like (a + b)(a - b) are unique products also known as the difference of squares.
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Answer the following questions:
1. a. 126 ÷ 12
b. 217 ÷ 4
c. 75 ÷ 6
d. 11 ÷ 9
e. 200 ÷ 35
2. a. Determine whether the quotient of 7 ÷ 2.2 is a terminating or nonterminating, repeating decimal.
b. Find out if the quotient of 0.5 ÷ 0.12 is a terminating or nonterminating, repeating decimal.
c. Which of these has a quotient that is a nonterminating, repeating decimal:
3 ÷ 0.8, 9 ÷ 0.16 or 2 ÷ 2.7?
d. Mr. Pelaez wants to divide his 1.3 hectares of land among his three children. How much land will each child get?
3. Identify if each resulting quotient is a terminating or nonterminating, repeating
decimal.
a. 2 ÷ 3
b. 5 ÷ 1.2
c. 0.3 ÷ 0.9
d. 2.4 ÷ 1.8
4. In a feeding activity, the teacher used 1.25 kilograms of ground pork to make 30 pieces of hamburger patties. How much meat was there in one piece of hamburger?
1. The quotients of the following division operations are as follows:
a) 126 ÷ 12 is 10.5.
b) 217 ÷ 4 is 54.25.
c) 75 ÷ 6 is 12.5.
d) 11 ÷ 9 is 1.2222.
e) 200 ÷ 35 is 5.7143.
2a) The quotient of 7 ÷ 2.2 is a nonterminating or repeating decimal.
2b) The quotient of 0.5 ÷ 0.12 is a nonterminating or repeating decimal.
2c) The division operation with a quotient that is nonterminating or repeating decimal is 2 ÷ 2.7.
3) The identification of each resulting quotient as a terminating or nonterminating, repeating decimal is as follows:
a. 2 ÷ 3 Nonterminating quotientb. 5 ÷ 1.2 Nonterminating quotientc. 0.3 ÷ 0.9 Nonterminating quotientd. 2.4 ÷ 1.8 Nonterminating quotient.4) The quantity of meat in one piece of hamburger, if they are evenly shared or divided, is 41.67 g.
What is the quotient?The quotient is the result of the division operation.
Every division operation involves the dividend (the numerator), the divisor (the denominator), the division operand (÷), and the equal sign (=).
2)
a) 7 ÷ 2.2 = 3.181818
b) 0.5 ÷ 0.12 = 4.16667
c) 3 ÷ 0.8 = 3.75
9 ÷ 0.16 = 56.25
2 ÷ 2.7 = 0.740740
3)
a. 2 ÷ 3 = 0.6667
b. 5 ÷ 1.2 = 4.16667
c. 0.3 ÷ 0.9 = 0.3333
d. 2.4 ÷ 1.8 = 1.3333
4) Total weight of ground pork = 1.25 kilograms
1 kg = 1,000 g
1.25 kg = 1,250 g
The pieces of patties = 30
The weight of each piece = 41.67 g (1,250/30)
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The diagram shows a cuboid with a volume of 42 cm³. K ? cm Work out the missing length of the cuboid. 2 cm 3 cm
Using the formula for the volume of cuboid that is the product of length, breadth and height, The answer is 7cm.
What is a cuboid?A cuboid is a three-dimensional shape with six rectangular faces, three of which are pairs of congruent squares and the other three are rectangles. It is also known as a rectangular prism. The opposite sides of a cuboid are parallel and equal in area. The length, width, and height of a cuboid are three of its measures of size.
What do you mean by volume and formula for volume of cuboid?The volume of a three-dimensional object is a measure of the amount of space it occupies. It is typically measured in units of cubic units, such as cubic centimeters (cm^3) or cubic meters (m^3). The formula for the volume of a cuboid is length x width x height. For example, if a cuboid has a length of 5 cm, a width of 3 cm, and a height of 2 cm, its volume would be 5 x 3 x 2 = 30 cubic centimeters. In general, the volume of a three-dimensional object can be found by multiplying the area of its base by its height.
length = 2cm
breadth =3cm
volume = 42 cm³
volume = l*b*h
42 = 2*3*h
h= 42/6 =7cm
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What is sin E for this triangle?
Enter your answer by filling in the boxes.
E
g
F
sin E =
I
e
Answer:
e/f
Step-by-step explanation:
sin =opposite/ hypothenuse
e is opposite
f is hypothenuse
With respect to the reference angle E, sinE = e/f.
What are trigonometric ratios in terms of a right-angle triangle?We know a right-angled triangle has three sides they are -: Hypotenuse,
Opposite and Adjacent.
We can remember SOH CAH TOA which is,
sin = opposite/hypotenuse, cos = adjecen/hypotenuse and
tan = opposite/adjacent.
We know, sin is opposite over hypotenuse.
From the given figure, With respect to the reference angle E,
opposite is FG and hypotenuse is EG.
Therefore, sinE = FG/EG.
sinE = e/f.
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Graph the line that passes through the points (-3, -1) and (1, -1) and
determine the equation of the line.
Answer:
y = -1
Step-by-step explanation:
The equation is y = mx + b
m = the slope
b = y-intercept
To find slope we look for the rise/run or (y2 - y1) / (x2 - x1)
The slope here is 0, and the y-intercept is located at (0,-1), so the equation is
y = -1
Answer:
y = -1
Step-by-step explanation:
Solve the systems algebraically please help :(
Answer:
I don't really know but I will help you if I can
Can a polynomial have no zero?
A quadratic polynomial does not cross the x axis at any point since it has no zeros.
An expression that consists of variables, constants, and exponents that is combined using mathematical operations like addition, subtraction, multiplication, and division is referred to as a polynomial (No division operation by a variable).
x^2 - 2x + 5 when compared with the standard form of the quadratic equation, a = 1, b=- 2, c=5
Value of the discriminant =b^2 - 4 ac
= (-2)2 - 4(1) (5)= 4 - 20
= - 15
The value of the discriminant is negative i.e less than zero.
Hence, the quadratic equation x^2 - 2x + 5 has no real zeroes.
Since a quadratic polynomial may only have two real roots, the number of times the graph will cross the x-axis at most is two, and at least zero when there are no roots in the equation.
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Find the product of 12 and 8. Find the greatest common factor of 12 and 8. Find the least common multiple of 12 and 8. Find the product of the greatest common factor and the least common multiple of 12 and 8.
Answer:
Product: 96
GCF: 4
LCM: 24
Product of the GCF and LCM: 96
Step-by-step explanation:
Product - The answer to a multiplication problem.GCF - The greatest positive integer that evenly divides both numbers.LCM - The smallest multiple that both numbers have in common.Now that we know what definitions are, we can solve this word problem.
12 × 8 = 96.
96 is the product.
To find the greatest common factor, we can list all of the factors for each number.
12: 1, 2, 3, 4, 6, 12
8: 1, 2, 4, 8
The largest factor both numbers have in common is 4.
Therefore, 4 is the greatest common factor.
To find the least common multiple, we can list the multiples of both numbers.
12: 12, 24, 36, 48, 60, 72, 84, 96
8: 8, 16, 24, 32, 40, 48, 56, 64
The smallest multiple both numbers have in common is 24.
Therefore, 24 is the least common multiple.
Lastly, we have to find the product of the GCF and the LCM.
The GCF is 4 and the LCM is 24.
4 × 24 = 96.
96 is the product.
Therefore, 96 is the product of the greatest common factor and the least common factor of 12 and 8.
What critical value of t* should be used for a 95% confidence interval for the population mean based on a random sample of 15 observations
A 95% confidence interval has a critical value of 1.96, where (1-0.95)/2 = 0.025. The unknown mean's 95% confidence interval is (101.82 - 0.96, 101.82 + 0.96), which translates to (101.82 - 0.96, 101.82 + 0.96) = (100.86, 102.78).
The critical t-value is what?The t-critical value marks the threshold at which the null hypothesis is accepted or disregarded. The null hypothesis is rejected if the t-statistic is closer to 0 than the t-critical value; otherwise, it is retained.
How do we determine the appropriate value of T *?Using a standard t-test and the following formula, you can determine a t-value: t is equal to (X - ц0) / (s / √n), where X is the sample mean, 0 is the population mean, s is the sample standard deviation, and n is the sample size.
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A 95% confidence interval has a critical value of 1.96, where (1-0.95)/2 = 0.025. The unknown mean's 95% confidence interval is (101.82 - 0.96, 101.82 + 0.96), which translates to (101.82 - 0.96, 101.82 + 0.96) = (100.86, 102.78).
The critical t-value is what?The t-critical value marks the threshold at which the null hypothesis is accepted or disregarded. The null hypothesis is rejected if the t-statistic is closer to 0 than the t-critical value; otherwise, it is retained.
How do we determine the appropriate value of T?Using a standard t-test and the following formula, you can determine a t-value: t is equal to [tex](X - ц0) / (s / √n)[/tex], where X is the sample mean, 0 is the population mean, s is the sample standard deviation, and n is the sample size.
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Based on the records for the past several seasons, a sports fan believes the probability the red team wins is . The fan also believes the probability the blue team wins is . In a season with games, how many games should the fan expect the red team to win?
See comment for complete question
Answer:
81 games
Step-by-step explanation:
Given
[tex]P(Red) = 0.45[/tex]
[tex]P(Blue) = 0.40[/tex]
[tex]n = 180[/tex]
Required
Expected game the red team will win
Expected number is calculated as:
[tex]E(x) = n * P(x)[/tex]
In this case, the number of winnings for the red team is:
[tex]Expected = 180 * P(Red)[/tex]
[tex]Expected = 180 * 0.45[/tex]
[tex]Expected = 81[/tex]
find the total surface area of this cuboid.
The total surface area of the cuboid is 2 (lh + lw+ hw) square units.
What is surface?A surface is the part of solid matter that can be touched by hand or seen with eyes. In other words, it is the place where the solid matter encounters the outside world through contact, light, etc. Solid matter not only effects humans with its surface, but also air, water, and other solid matter. Therefore, things around us work cooperatively with the outside world and with other things through surfaces, without a surface, solid matter would not be able to interact with anything.By changing the characteristics of solid matter, interactions such as the impact on the solid matter from other things (such as environmental conditions and people) as well as the effect that solid matter has on other things can be changed. In other words, certain features of the solid matter can be controlled as needed.To learn more about surface refer to:
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hi plz answer image below may god less you no bot answers plz
Answer:42.875 feet3
Step-by-step explanation:
Answer:
Step-by-step explanation:
V=a³
=[tex](3\frac{1}{2})^3=42\frac{7}{8}[/tex]in³
What is line segment Theorem?
Line segment theorem states that if a line segment is bisected by another line, then the two parts of the line segment are equal.
This theorem is also known as the Midpoint Theorem. The formula for the Line Segment Theorem is A/B = C/D, where A and C are the two halves of the line segment, and B and D are the distances from the midpoint to the two endpoints of the line segment.
For example, consider a line segment AB with endpoints at A(-3, 2) and B(3, 2). The midpoint of the line segment is at (0,2). To calculate the length of each half of the line segment, we would use the distance formula and plug in the coordinates of the two endpoints and the midpoint.
For half A, the distance is d(A, M) = sqrt[(-3 - 0)^2 + (2 -2)^2] = 3.
For half C, the distance is d(M, B) = sqrt[(3 - 0)^2 + (2 -2)^2] = 3.
Therefore, A/B = 3/3 = 1 and C/D = 3/3 = 1. This confirms that the two halves of the line segment are equal in length.
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9 Jugadores de un equipo de baloncesto chocan palmadas unos con otros ¿Cuántas palmadas dan en total? 2 jugadores dan una palmada. 3 jugadores, tres palmadas. 4 jugadores, seis palmadas. 5 jugadores, diez palmadas.
Answer:
Los jugadores dan 36 palmadas en total.
Step-by-step explanation:
Dado que 9 jugadores de un equipo de baloncesto chocan palmadas unos con otros, para determinar cuántas palmadas dan en total, sabiendo que 2 jugadores dan una palmada, 3 jugadores, tres palmadas, 4 jugadores, seis palmadas, y 5 jugadores, diez palmadas, se debe realizar el siguiente cálculo:
Cada jugador da un número de palmadas igual al número de jugadores en el campo, menos 1. A su vez, esa palmada es recíproca con el otro jugador que la recibe, con lo cual el número de jugadores a contar va disminuyendo.
8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = X
15 + 6 + 5 + 4 + 3 + 2 + 1 = X
21 + 5 + 4 + 3 + 2 + 1 = X
26 + 4 + 3 + 2 + 1 = X
30 + 3 + 2 + 1 = X
33 + 2 + 1 = X
35 + 1 = X
36 = X
Por lo tanto, dan 36 palmadas en total.
What is the 3 variable?
A three-variable system is a system of equations or inequalities involving three variables.
The three variables could refer to any three related quantities, such as a person's age, height, and weight. In a three-variable system, the number of equations is equal to the number of variables, so three equations are needed to solve the system. Each equation in the system will involve all three variables and will be solved through a combination of algebraic manipulation, substitution, and elimination techniques. Once all three equations have been solved, the values of the variables can be found.
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my helpers help me again -^-
Answer:
g= 92°
h= 88°
k= 91°
m= 89°
Step-by-step explanation:
Answer:
g = 92°
h = 88°
k = 91°
m = 89°
Step-by-step explanation:
Have a nice day!