Logarithm is a mathematical operation that is the inverse of exponentiation. It is denoted by log b (x), which means the logarithm of x to base b.
For example, log2 8 = 3, since 23 = 8. Logarithm is useful in solving equations that involve exponential terms.
The properties of logarithm are:
1. log b 1 = 0: This property states that the logarithm of 1 is zero for any base b.
2. log b b = 1: This property states that the logarithm of any base b is 1.
3. log b (xy) = log b (x) + log b (y): This property states that the logarithm of the product of two numbers is equal to the sum of the logarithms of the two numbers. For example, log2 (8x32) = log2 8 + log2 32 = 3 + 5 = 8.
4. log b (x/y) = log b (x) - log b (y): This property states that the logarithm of the quotient of two numbers is equal to the difference of the logarithms of the two numbers. For example, log2 (8/32) = log2 8 - log2 32 = 3 - 5 = -2.
5. log b c = clog b (c): This property states that the logarithm of a number c to the base b is equal to c times the logarithm of b. For example, log2 32 = 32log2 2 = 5.
These properties of logarithm are very useful in solving mathematical problems involving exponential terms.
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The volume of a box is 231 cubic inches. The width is 7 inches. The length is 8 inches more than the height. Find the missing dimensions.
Answer:The missing dimensions are now
width =7inches
length = 11 inches
Height = 3 inches
Step-by-step explanation:
Volume of a box is given as =width x length x height 231 in^3
Given
width = 7 inches
height =represented as h
l = 8 inches more than the height= 8+h
V = w x l x h
231 = 7(8+h)(h)
231=( h² + 8h)7
231/ 7= h²+8h
33= h²+8h
Giving us the quadratic equation
h² + 8h - 33 = 0
=(h-3)(h+11) = 0
h-3= 0
h= 3 also h+11=0, h=-11
h= 3 or -11
Since the value of height cannot have a negative value,
h = 3inch
l = 8+h = 8+3 = 11 in
The missing dimensions are now
width =7inches
length = 11 inches
Height = 3 inches
Please help due by tonight.
( . ) ( . )
\
\____/
Answer:
y = 1/2x^2 - 2x - 1
Step-by-step explanation:
Vertex(2, - 3)
Equation of Parabola (In vertex form): y = a(x - h)^2 + k,
y = a(x - 2)^2 - 3
Substitute given point (0, -1) and solve for a:
y = a(x - 2)^2 - 3,
-1 = a(0 - 2)^2 - 3,
a = 1/2
Equation: y = 1/2(x - 2)^2 - 3 = 1/2(x^2 - 4x + 4) - 3 = 1/2x^2 - 2x + 2 - 3 = 1/2x^2 - 2x - 1,
y = 1/2x^2 - 2x - 1: Option C
A rectangle measures 2 2/3
inches by 2 1/3 inches. What is its area?
thank you!
Answer:
Step-by-step explanation:
To find the area of a rectangle, we multiply its length by its width.
The length of the rectangle is 2 2/3 inches, which can be expressed as an improper fraction: (3 * 2 + 2)/3 = 8/3 inches.
The width of the rectangle is 2 1/3 inches, which can also be expressed as an improper fraction: (3 * 2 + 1)/3 = 7/3 inches.
Now, we can calculate the area by multiplying the length and width:
Area = (8/3) * (7/3)
= (8 * 7)/(3 * 3)
= 56/9
Therefore, the area of the rectangle is 56/9 square inches, which can be simplified, if needed.
What is the solution to the systems of linear equations shown?
Answer:
C
Step-by-step explanation:
The solution to the systems of linear equations through graphing is simply the coordinates of the intersection point of the two lines.
The coordinates of the intersection line is written like this: (x, y)
Since the point is 0 on the x-axis and 3 on the y-axis, the Solution = (0, 3)
can answer this please
Answer:
[tex] \frac{a}{b} = \frac{3}{2} [/tex]
Step-by-step explanation:
Since both the triangles are similar, so their corresponding sides would be in proportion.
Therefore,
[tex] \frac{a}{b} = \frac{2.1}{1.4} \\ \\ \frac{a}{b} = \frac{3}{2} [/tex]
Solve for n. N3 = 16
The heights of a certain breed of dogs has a normal distribution with a mean of 28 inches and a standard deviation of 4 inches. If we randomly select 64 of these dogs, what is the probability that the mean height of 64 dogs is: a) Less than 27 inches? b) Greater than 28.5 inches? c) Between 27 and 28.5 inches?
The probability that the mean height of 64 dogs is between 27 and 28.5 inches is approximately 0.8531.
We can use the central limit theorem to approximate the distribution of the sample mean. The central limit theorem states that if we take a large enough sample from a population, the sample mean will be approximately normally distributed with mean equal to the population mean and standard deviation equal to the population standard deviation divided by the square root of the sample size. In this case, we have:
Population mean (μ) = 28 inches
Population standard deviation (σ) = 4 inches
Sample size (n) = 64
a) To find the probability that the mean height of 64 dogs is less than 27 inches, we need to standardize the sample mean and find the corresponding area under the standard normal distribution. We have:
z = (sample mean - population mean) / (population standard deviation / sqrt(sample size))
z = (27 - 28) / (4 / sqrt(64))
z = -2
Using a standard normal distribution table or calculator, we find that the probability of z being less than -2 is approximately 0.0228. Therefore, the probability that the mean height of 64 dogs is less than 27 inches is approximately 0.0228.
b) To find the probability that the mean height of 64 dogs is greater than 28.5 inches, we standardize the sample mean and find the area to the right of the standardized value. We have:
z = (sample mean - population mean) / (population standard deviation / sqrt(sample size))
z = (28.5 - 28) / (4 / sqrt(64))
z = 1
Using a standard normal distribution table or calculator, we find that the probability of z being greater than 1 is approximately 0.1587. Therefore, the probability that the mean height of 64 dogs is greater than 28.5 inches is approximately 0.1587.
c) To find the probability that the mean height of 64 dogs is between 27 and 28.5 inches, we need to find the area under the standard normal distribution between the two standardized values. We have:
z1 = (27 - 28) / (4 / sqrt(64))
z1 = -2
z2 = (28.5 - 28) / (4 / sqrt(64))
z2 = 1
Using a standard normal distribution table or calculator, we find that the probability of z being between -2 and 1 is approximately 0.8531. Therefore, the probability that the mean height of 64 dogs is between 27 and 28.5 inches is approximately 0.8531.
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PLS HELP!! BRAINLIESt & 20 POINTS!! AND AN EXTRA THANKS AND 5 STARS
You survey your class to find out the types of pets they have at home.
Below are the results:
Dog, Dog, Cat, Cat, Cat, Dog, Hamster, Fish, Dog, Fish
a. Make a frequency table for the results. Continue on the following page.
b. Calculate the relative frequencies of the each animal type.
c. Using the relative frequencies explain which animals are most and least popular. Be specific and explain your reasoning.
Answer:
Step-by-step explanation:
a) Frequency table:
Category Frequency
Dog 4
Cat 3
Fish 2
Hamster 1
b) Relative frequencies of each animal type
Dog: 4/10 = 0.4
Cat: 3/10 = 0.3
Fish: 2/10 = 0.2
Hamster: 1/10 = 0.1
c) Popularity
Dog is the most popular because it has the highest relative frequency.
Hamster is the least popular because it has the lowest relative frequency.
Natalie is a lab technician and earns $10.65 per hour. What are her wages for 40 hours?
pls help im in need
Samples of size 60 are drawn from a large population known to have a mean of 78 and a standard deviation of 12. What proportion of these samples would we expect to have a mean greater than 80?
a
0.566
b
0.098
c
Almost none
d
0.434
e
0.901
Answer: 0.098
Step-by-step explanation: First you’ll need to find the sample distribution mean and standard deviation (which is 78 and 1.55). Then you plug it into calculator using normal approximation cause n is large: normalcdf(80,100,78,1.55)
As, 0.098 proportion of these samples would, we expect to have a mean greater than 80. Thus, option (C) is correct.
What is the population?As the population is defined, as a group of groupings with a common characteristic. In data point, a population is the pool of individuals from which a statistical sample is drawn for a study. As we see, different countries have different populations.
As here the μ = 78, σ = 1.5492 and x = 80
We need to compute P(X >= 80). The corresponding z-value is calculated using Central Limit Theorem
z = (x - μ)/σ
z = (80 - 78)/1.5492 = 1.29
Therefore,
P(X >= 80) = P(z <= (80 - 78)/1.5492)
= P(z >= 1.29)
= 1 - 0.902 = 0.098
Therefore, The right option (B) is correct
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Un cliente tiene que devolver el dinero al banco donde solicitó un préstamo de $8 000.00 el 27 de diciembre. Si fue devuelto el 27 de octubre de ese mismo año y la tasa de descuento aplicada fue del 3% anual. ¿Cuál es el descuento que corresponde a esta operación? ¿Cuánto tiene qué pagar el cliente el 27 de octubre?
Answer:
694
Step-by-step explanation:
2 Simplify -3(16 = 35+4)/5. Show your work.
Answer:
-9
Step-by-step explanation:
Parentheses first:16-35+4 = -35+16+4 = -35+20 = -15
3(-15) = -45/5
-45 divided by 5 = -9
help meee plssssssss
Answer:
A) Triangle PQR
Step-by-step explanation:
This is the correct triangle because it has the exact same measurements as Triangle ABC. Angles A and P are the same angle on the triangles and they have the same measure (70). Angles C and R are the same angle on the triangles and they have the same measure (75). To solve for Angle C in Triangle ABC though, add angles 70 + 35 together to get 105. A triangle has a total measure of 180, so subtract 180 - 105 to get 75.
Hope it helps!
12. the number of errors in a textbook follows a poisson distribution with a mean of 0.04 errors per page. what is the expected number of errors in a textbook that has 204 pages? circle one answer.
The number of errors in a textbook follows a Poisson distribution with a mean of 0.04 errors per page. To find the expected number of errors in a textbook with 204 pages, we need to multiply the mean by the number of pages.
Expected number of errors = mean * number of pages = 0.04 * 204 = 8.16
Therefore, we can expect to find approximately 8 errors in a textbook that has 204 pages, based on the given Poisson distribution with a mean of 0.04 errors per page. It is important to note that this is only an expected value and the actual number of errors could vary.
Additionally, Poisson distribution assumes that the errors occur independently and at a constant rate, which may not always be the case in reality. Nonetheless, the Poisson distribution provides a useful approximation for the expected number of rare events occurring in a given interval.
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Please help meeeeeee 67 points
Answer:
B.) 20 degrees
Step-by-step explanation:
We know that the sum of angles in a triangle is equal to 180, so we can say that 80 + 4x + x is equal to 180. This means that 5x is 100 degrees, meaning that x = 20 degrees. The option with this answer is B.).
Answer:
The answer is B.
Step-by-step explanation:
4 × 20 = 80
80 + 80 = 160
A triangle is 180°
Find the missing x.
180 - 160 = 20
Plug 20 in for both x's to check your work.
Hope this helps :)
-ilovejiminssi♡
the sampling distribution of is normal if the sampled populations are normal, and approximately normal if the populations are nonnormal and the sample sizes n1 and n2 are large. a. true b. false
The sampling distribution is normal if the sampled populations are normal, and approximately normal if the populations are nonnormal and the sample sizes n1 and n2 are large: (A) TRUE
The central limit theorem states that as sample sizes increase, the distribution of the sample means approaches a normal distribution regardless of the shape of the population distribution, as long as the samples are randomly selected and independent.
Therefore, if the populations from which the samples are drawn are normal, the sampling distribution of the means will also be normal.
However, even if the populations are nonnormal, the sampling distribution will still be approximately normal if the sample sizes are large enough.
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let a be an n x n matrix with an eigenvalue of multiplicity n. show that a is diagonalizable if and only if a = i
An n x n matrix a with an eigenvalue of multiplicity n is diagonalizable if and only if a = i, where i is the identity matrix.
Suppose a is diagonalizable. Then there exists an invertible matrix P such that a = PDP^(-1), where D is a diagonal matrix. Since a has an eigenvalue of multiplicity n, the diagonal entries of D are all equal to that eigenvalue. Therefore, a = PDP^(-1) = P(lambda I)P^(-1) for some scalar lambda. But since the eigenvalue has multiplicity n, lambda must equal the eigenvalue, which implies that D = lambda I. Therefore, a = [tex]P(lambda I)P^(-1) = PDP^(-1)[/tex] = P(lambda I)P^(-1) = lambda PPP^(-1) = lambda I. Thus, a = lambda I, and since the eigenvalue has multiplicity n, we have lambda = 1. Therefore, a = i.
Conversely, suppose a = i. Then a is trivially diagonalizable, since any matrix is diagonalizable if and only if it is already diagonal. Therefore, a is diagonalizable, and the proof is complete.
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Consider the following integral. Sketch its region of integration in the xy-plane.
∫1_0∫y_(√y) 110x2y3dxdy
(a) Which graph shows the region of integration in the xy-plane? ? A B
(b) Evaluate the integral.
a. The region of integration is the region between the curves[tex]y = x^4[/tex]and [tex]y = x^2[/tex], for 0 ≤ y ≤ 1.
b. The value of the integral is -0.6875.
To sketch the region of integration, we need to find the boundaries of the integral.
We are integrating with respect to x first, so we need to express the limits of integration for x in terms of y.
From the inner integral, we have:
y ≤ [tex]x^2[/tex] ≤ √y
Taking the square root of both sides of the right inequality, we get:
[tex]y^{1/4}[/tex] ≤ x ≤ √y
Thus, the region of integration is the region between the curves[tex]y = x^4[/tex]and [tex]y = x^2[/tex], for 0 ≤ y ≤ 1.
To sketch the region, we can first draw the curves [tex]y = x^4[/tex] and [tex]y = x^2,[/tex]as shown in graph (A) below:
| .
1 | .
| .
| .
*------------>
0 1 2 3
The region of integration is the shaded region between the curves, as shown in graph (B) below:
| +
1 | +
| +++
| +++
*------------>
0 1 2 3
To evaluate the integral, we can use the boundaries we found and evaluate the integral using the following steps:
[tex]\int 1_0 \int y_(\sqrt{y } ) 110x^2y^3dxdy[/tex]
[tex]= \int 0^1 \int x^{1/2}^x^2 110x^2y^3 dy dx[/tex] (using the limits we found earlier)
[tex]= \int 0^1 110x^2 [(1/4)y^4]_(x^{1/2})^{x^2} dx[/tex] (integrating with respect to y)
[tex]= \int 0^1 110x^2 [(1/4)x^8 - (1/4)x^4] dx[/tex] (substituting the limits of integration)
[tex]= \int 0^1 (27.5x^10 - 27.5x^6) dx[/tex].
[tex]= [2.75x^11 - 3.4375x^7]_0^1[/tex](integrating and substituting limits)
= 2.75 - 3.4375.
= -0.6875.
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Duane And his family are playing a game Dwayne scores 11 points each tile matches 1/3 point model the solution with an equation how many tile matches this Dwayne make
Answer:
33 tiles
Step-by-step explanation:
Given
[tex]1\ tile = \frac{1}{3}\ points[/tex]
Required
Determine the number of tiles in 11 points
Represent this with x. So, we have:
[tex]1\ tile = \frac{1}{3}\ points[/tex]
[tex]x = 11\ points[/tex]
Cross Multiply
[tex]x * \frac{1}{3} = 11 * 1[/tex]
Multiply through by 3
[tex]3*x * \frac{1}{3} = 11 * 1*3[/tex]
[tex]x = 33[/tex]
what is the y value when x= -1?
what is the y value when x=0?
what is the y value when x=2?
Answer:
-1, y = 2
0, y = 3
2, y = 5
Step-by-step explanation:
-1, y = -1 + 3
0, y = 0 + 3
2, y = 2 + 3
50 POINTS!
In the diagram, d = 4. What is the value of x? Express your solution in simplest radical form, if necessary.
Answer:
146
Step-by-step explanation:
180=4+30+x
180=34+x
180-34=x
146=x
Select the values of that make the inequality true.
X< 2.1
Answer:
B, C, and D
Step-by-step explanation:
They all have lower values than 2.1
Find an ordered pair (x, y) that is a solution to the equation. x-3y = 6
Answer:
(0, -2)
(6, 0)
Step-by-step explanation:
pick a value for 'x', plug it into the equation, then solve for 'y'
pick a value for 'y', plug it into the equation, then solve for 'x'
Can someone tell me the answer
Answer:
y=8x+4
Step-by-step explanation:
A cookie jar starts off with 32 cookies in it, and
each day 2 cookies are eaten. After a certain
number of days, there are 14 cookies left in the jar.
Which of the following equations represents the
day, d, when there are 14 cookies in the jar?
Answer:
9 days
Step-by-step explanation:
32-2-2-2-2-2-2-2-2-2 = 14
The equation that represents the day, d, when there are 14 cookies in the jar will be 14 = 32 - 2d, and that day will be on the 9th day.
What is the equation?There are many different ways to define an equation. The definition of an equation in algebra is a mathematical statement that demonstrates the equality of 2 mathematical expressions.
As per the given,
Initial cookie = 32
Since each day 2 cookies are eaten so after "d" days the total eaten cookies will be 2d.
Remaining cookies = initial cookies - eaten cookies
Remaining cookies = 32 - 2d
The day when 14 cookies remain will be,
14 = 32 - 2d
2d = 32 - 14
d = 9
Hence "The equation that represents the day, d, when there are 14 cookies in the jar will be 14 = 32 - 2d, and that day will be on the 9th day".
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State if the triangles are similar. If so, state how you know they are.
Answer:
The triangles are similar
Step-by-step explanation:
Since EF ║ UT, m∠FUT = m∠EFU because alternate interior angles are congruent when lines are parallel.
m∠UDT = m∠EDF because vertical angles are congruetnt
Therefore ΔUDF is similar to ΔFDE because if two angles of a triangle are congruent to two angles of another triangle, then the triangles are similar.
Suppose an upward force of 15 N is added to the box. What will be the net vertical force on the box?
Answer:
Step-by-step explanation:
Suppose an upward force of 15 N is added to the box. What will be the net vertical force on the box?
THIS IS THE COMPLETE QUESTION BELOW;
1. Explain how you calculate the net force in any direction on the box.
2. Suppose an upward force of 15 N is added to the box. What will be the net vertical force on the box?
3. What force could be applied to the box to make the net force in the horizontal direction zero? Explain.
4. Suppose a force of 25 N to the right is added to the box. What will be the net force to the right?
CHECK THE ATTACHMENT FOR THE FIQURE.
1)since the given horizontal forces are acting on the box at opposite direction then
NET FORCE= (100- 50)N
= 50N( this is because the 100N and 50N acting Horizontally will cancel each other, then the remaining force is 50N.
✓Since, the two vertical forces also acted on the box in opposite direction, then
NET FORCE= (25 25)N
= 0N
Note: horizontal forces are one acting from right to left and vice versa, while vertical forces are ones acting from up to down and vice versa.
HENCE, NET FORCE = 50N + 0N= 50N
2) since the net vertical force was 0N, if
15N force is added then we still have NET FORCE OF 15N
3) if force of 50N is applied to horizontal forces moving from left to right then NET HORIZONTAL FORCES= 0N
4) since, the net force for the box is 50N, if Horizontal force of 25N is added moving left to right then NET FORCE = 25N for whole box.
what is speaker? what is printer
Answer:
A speaker is a term used to describe the user who is giving vocal commands to a software program. ... A computer speaker is an output hardware device that connects to a computer to generate sound. The signal used to produce the sound that comes from a computer speaker is created by the computer's sound card
Step-by-step explanation:
rate nyoooo plssss
pleasee helpp honestly im stuck i dont know how to graph this equation. it says to graph the funcion c(x)=3x-1
first take the 3 and place it over the negative one in quadrant 2
Answer:
apply number to the X when x= y = 3*0-1 =-1
x=1 y = 3*1-1 = 2
x = 2 y = 3*2-1 =5
what's
[tex]1024 \div 256[/tex]
Answer:
4 ...................
Step-by-step explanation:
hope the answer was helpful ......
Answer:
1024÷256=4
Step-by-step explanation:
hope u get that