suppose that a > 1 and that f (n) = (loga n). show that f (n) = (lg n).

Answers

Answer 1

Given that a > 1 and f(n) = log_a(n), you want to show that f(n) = lg(n). loga n is equivalent to lg n when a > 1.

First, let's clarify that "lg" typically represents the base-10 logarithm, i.e., log_10(n). Now, we can use the change of base formula for logarithms to establish the relationship between log_a(n) and lg(n):

f(n) = log_a(n) = log_10(n) / log_10(a) = lg(n) / lg(a).

To show that f(n) = lg(n), we need to establish that lg(a) = 1. However, this is not generally true for all a > 1. Therefore, we cannot conclude that f(n) = lg(n) for all cases. The relationship between f(n) and lg(n) is given by the equation f(n) = lg(n) / lg(a).

To show that f(n) = (lg n), we need to prove that loga n is equivalent to lg n.

First, let's recall the change of base formula:

logb x = loga x / loga b

We can use this formula to rewrite loga n as:

loga n = log2 n / log2 a

Now, let's substitute a = 2 in the above equation:

log2 n = log2 n / log2 2

Since log2 2 = 1, we can simplify the equation to:

log2 n = log2 n

This proves that f(n) = (lg n). Therefore, loga n is equivalent to lg n when a > 1.

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Related Questions

how many ozens in a gallon

Answers

Twelve dozen is referred to as a gross (122 = 144) and twelve gross (123 = 1,728, the duodecimal 1,000) is referred to as a great gross, a word most frequently used when shipping or purchasing products in bulk. The equivalent of 120 or ten dozen is a great hundred, commonly referred to as a small gross.

What is Dozen?A group of twelve is referred to as a dozen and is frequently abbreviated as doz or dz.Given that there are roughly twelve lunar cycles (or months) in each cycle of the sun (or year), the number twelve may be one of the first primitive integer groupings. Twelve is useful because it has the greatest number of divisors among numbers up to its double, a quality shared only by the numbers 1, 2, 6, 12, 60, 360, and 2520. The duodecimal system, commonly referred to as dozenal, was developed in Mesopotamia and uses twelve as its base number (see also sexagesimal). On the hands, numbering each finger bone with the thumb makes it simple to count in base-12.By using the second hand as a stand-in for units of twelve, two hands may count to 144 using this system, which allows one hand to count to twelve. Twelve dozen is referred to as a gross (122 = 144) and twelve gross (123 = 1,728, the duodecimal 1,000) is referred to as a great gross, a word most frequently used when shipping or purchasing products in bulk. The equivalent of 120 or ten dozen is a great hundred, commonly referred to as a small gross.Some products, which can vary by nation and are frequently foodstuffs, are packaged or sold by the dozen (a dozen eggs). One source claims that sales by the dozen may have its origins in English units, a system where "one egg may be sold for a penny or 12 for a shilling, which was the standard price for eggs at the time."

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A family took a trip. They spent $71.90 for gasoline, $135.00 on meals, $179.00 on amusements, and $37.83 on miscellaneous expenses. How much did the trip cost?

Answers

Answer:

423.73

Step-by-step explanation

If you add all those together the answer would be 423.7

For a particular model of watch, consumers will demand 92 items when the price is $231, and demand 228 items when the price is reduced to $129. For the same watch, suppliers are willing to sell 54 watches when the price is $102, and 258 watches when the price is $204. Give the linear equation, in slope intercept fo, for the demand of this product. Remember, we want the fo p = mq+b. Reduce your slope to simplest tes as a fraction or as a decimal to two places of accuracy.

Answers

The slope is 4/3, and the y-intercept is 49.33 of liner equation.

The given demand and price values are shown below:

Price, p: 231, 129

Demand, q: 92, 228

From the given data we can find the slope and y-intercept to find the linear equation in the slope-intercept form.

Linear equation in slope-intercept form is y = mx + b.

where

y is the dependent variable,

x is the independent variable,

m is the slope and b is the y-intercept.

The slope is defined as the change in the dependent variable divided by the change in the independent variable.

m = (change in demand/change in price)

By using the above formula we can find the slope of the linear equation. Let's substitute the values in the above formula.

m = (228 - 92)/(129 - 231)m = -136/-102m = 1.3333 or 4/3

Now, we know the slope of the linear equation. Next, let's find the y-intercept.

The linear equation is represented as p = mq + b,

where p is the price,

m is the slope,

q is the quantity demanded, and

b is the y-intercept.

To find the value of b, let's substitute the slope value and any of the above-given points.

Let's take (231, 92).p = mq + b231 = 4/3(92) + bb = 231 - 4/3(92)b = 49.33

Thus the equation for the demand for this product is: p = mq + bp = (4/3)q + 49.33

The slope is 4/3, and the y-intercept is 49.33.

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Which function has an inverse that is a function?

Answers

Answer: C

Step-by-step explanation:

On the attached diagram you can see all graphs of functions b(x), d(x), m(x) and p(x).

Finding the inverse of a function f(x):

1. First, replace f(x) with y. This is done to make the rest of the process easier.

2. Replace every x with a y and replace every y with an x.

3. Solve the equation from Step 2 for y. This is the step where mistakes are most often made so be careful with this step.

4. Replace y with In other words, you’ve managed to find the inverse.

5. Remember: the domain of f is the range of and the range of f is the domain of .

Using this algorithm, you can find the inverse only in case C:

for m(x)=-7x:

1. y=-7x.

2. x=-7y.

3. y=-x/7.

4. .

5. The domain and the range of m(x) are all real numbers as well as the domain and the range of

Functions b(x) and p(x) are not one-to-one functions (see attached diagram), then you can't find an inverse function. Function d(x) doesn't include x, then you can't also find an inverse function.

What is 127 lbs to kg

Answers

57.594 kilos are equal to 127 pounds. Divide the weight in pounds by 2.2 to get the equivalent weight in kilogram .

1 lb is equal to 0.45359237 kg.

127 lbs is equal to 127 x 0.45359237 kg, or 57.594 kg.

57.594 kilos are equal to 127 pounds. Divide the weight in pounds by 2.2 to get the equivalent weight in kilogrammes. Given that one pound is equivalent to 0.45359237 kilogrammes, this is true. Thus, multiply the weight in pounds by 0.45359237 to get the precise number of kilos.Let's use 127 pounds as an example. Multiply this by 0.45359237 to convert to kilos. 57.594 kg is the outcome, according to this. This is precisely how many kilogram are needed to weigh 127 pounds.Pounds to kilogram translation is a straightforward operation. All you have to do is multiply 0.45359237 by the weight in pounds after dividing it by 2.2. You will receive the kilogramme result for the specified weight in pounds as a result. It's vital to remember that only weights measured in pounds and kilogrammes can be converted using this method. Other computations may be needed if you want to convert between different measuring units.

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The town of Manchester contains a business area in the center of town that has the shape of a disk with a radius of 11 mile.

If no one resides in the business area, what is the population density in Manchester, in people per square mile, outside of the

business area?

Round your answer to the nearest whole number of people per square mile.

Answers

Answer:

58,241

Step-by-step explanation:

I just searched it up and it said the answer!

The ______ ________, denoted (P Hat), is given by the formula P= X/N, where X is the number of individuals with a specified characteristic in a sample of N individuals.

Answers

The sample proportions denoted p is given by the formula p =x/n, where x is the number of individuals with a specified character in a sample of n individuals.

In statistics, a Sample proportion, often denoted by P or the Greek letter pi, is a parameter that describes a percentage value related to a population. For example, the 2010 US Census showed that 83.7% of the US population was neither Hispanic nor Latino; the value of 0.837 is the population ratio. In general, the population proportion and other population parameters are unknown. A census can be performed to determine the true value of a population parameter, but it is usually impractical due to cost and time consumption.

Population proportions are usually estimated from unbiased sample statistics obtained from studies or observational experiments. For example, the National Technology Literacy Conference conducted a nationwide survey of 2,000 adults to determine the percentage of economically illiterate adults. Research shows that 72% of the 2,000 adults surveyed did not understand what Gross Domestic Product is. The value of 72% is a sample proportion. The sample proportion is usually denoted p and is denoted p in some textbooks.

According to the Question:

From the text, we have the following highlights.

The parameter involves x and n

n represent the sample, while x is a subset of n.

The parameter is denoted by p

In statistics, the parameter denoted by p is the sample proportion.

And it is calculated using:

P = x/n

Hence, the missing texts are: sample proportion and x/n.

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The electric potential of a point charge Q located at the origin of the Cartesian coordinate system is V=
4πε
0

(x
2
+y
2
+z
2
)
1/2

Q

. Find the corresponding electric field E. (a) E=
4πε
0

Q

(x
2
+y
2
+z
2
)
1/2

x
x
^
+y
y
^

+z
z
^

(b) E=
4πε
0

Q

(x
2
+y
2
+z
2
)
2

x
x
^
+y
y
^

+z
z
^

(c) E=
4πε
0

Q

(x
2
+y
2
+z
2
)
x
x
^
+y
y
^

+z
z
^

(d) E=
4πε
0

Q

(x
2
+y
2
+z
2
)
3/2

x
x
^
+y
y
^

+z
z
^

Answers

The correct option that represents the electric field is:

Option D: E = 4πε₀Q(x² + y² + z²)^(3/2)x^ + y^ + z^

How to find the Electric Field using partial Derivatives?

The electric field is given by the formula:

E = -∇V

where:

∇ is the del operator, which in Cartesian coordinates is represented as:

∇ = (∂/∂x)x^ + (∂/∂y)y^ + (∂/∂z)z^

Taking the negative gradient of V gives us:

E = -∇V = -[(∂V/∂x)x^ + (∂V/∂y)y^ + (∂V/∂z)z^]

First, let's calculate the partial derivatives of V with respect to x, y, and z:

(∂V/∂x) = (∂/∂x)[4πε₀(Q/√(x² + y² + z²))]

= -4πε₀(Qx/√(x² + y² + z²)³)

(∂V/∂y) = (∂/∂y) [4πε₀(Q/√(x² + y² + z²))]

= -4πε₀(Qy/√(x² + y² + z²)³)

(∂V/∂z) = (∂/∂z) [4πε₀(Q/√(x² + y² + z²))]

= -4πε₀(Qz/√(x² + y² + z²)³)

Now, substituting these derivatives into the expression for E:

E = -[(-4πε₀(Qx/√(x² + y² + z²)³))x^ + (-4πε₀(Qy/√(x² + y² + z²)³))y^ + (-4πε₀(Qz/√(x² + y² + z²)³))z^]

Simplifying, we have:

E = 4πε₀Q(x/√(x² + y² + z²)³)x^ + 4πε₀Q(y/√(x² + y² + z²)³)y^ + 4πε₀Q(z/√(x² + y² + z²)³) z^

Therefore, the correct option is:

(d) E = 4πε₀Q(x² + y² + z²)^(3/2)x^ + y^ + z^

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Find measure of angle E

Answers

The measure of ∠E will be equal to 38°.

Given,

m∠BFE = 118° and m∠C = 86°

And BF bisects ∠ABD

From the Property of Linear Pair,

∠AFB + ∠BFE = 180°

=> ∠AFB + 118° = 180°

=> ∠AFB = 62°

Now, we can see that ∠AFB and ∠FBD are equal because they are alternate angles.

And ∠FBD and ∠ABF are equal because BF bisects ∠ABD.

So, In ∆ABF,

∠AFB + ∠ABF + ∠BAF = 180° (because of the angle sum property of a triangle)

=>62° + 62° + ∠BAF = 180°

=>∠BAF = 180° - 124°

=> ∠BAF = 56°

Similarly, in ∆ABE,

∠A +∠C +∠E = 180°

=> 56° + 86° + ∠E = 180°

=> ∠E = 38°

Therefore, m∠E = 38°.

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Determine if the product of 3/2and 8/18 is rational or irrational. Explain your answer.

Answers

Answer:

Step-by-step explanation:

Please answer thank you so much

Answers

Answer:

2.85 and 3.45

Step-by-step explanation:

it gpea by 20s I think

I really need help on this

Answers

i tried to be as detailed as possible...if you’re mainly just looking for answers...look for the arrows like this ↳

Please help !!!!! Don’t answer if you just want to scribble . Imagine how you would feel if it were you .

Answers

Answer:

m= -3

b= 4

Step-by-step explanation:

whatever number has x next to it is always slope, that is how you find slope in a equation. To find y-intercept (b) you can just use the other number that doesn't have x to it.

Hope this helps!!!

For each integral, indicate what trig function would be used to solve using trigonometric substitution. (No work to show.) a) ∫ x 2 +4 x 2 dx b) ∫ 16−9x 2 x 2 dx c) ∫ 5x 2 −3 x 2 dx

Answers

As per the trigonometric substitution the value of the integrals are

a) ∫ x² + 4 / x² dx is 2 tan θ

b) ∫ (16 - 9x²) / x² dx is (4/3) sec θ

c) ∫ (5x² - 3) / x² dx is (1/√3)tanθ

Let's take a look at the given integrals:

a) ∫ x² + 4 / x² dx

For this integral, we can use the trigonometric substitution x = 2tanθ. This substitution simplifies the integral and allows us to solve it using trigonometric identities.

b) ∫ (16 - 9x²) / x² dx

To solve this integral, we can use the trigonometric substitution x = (4/3)secθ. This substitution simplifies the expression under the integral sign and allows us to use trigonometric identities to solve the integral.

c) ∫ (5x² - 3) / x² dx

For this integral, we can use the trigonometric substitution x = (1/√3)tanθ. This substitution simplifies the integral and allows us to use trigonometric identities to solve it.

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Please help me with edge question .

Answers

$~\hspace{7em}\textit{negative exponents} \\\\ a^{-n} \implies \cfrac{1}{a^n} ~\hspace{4.5em} a^n\implies \cfrac{1}{a^{-n}} ~\hspace{4.5em} \cfrac{a^n}{a^m}\implies a^na^{-m}\implies a^{n-m} \\\\[-0.35em] ~\dotfill\\\\ (4)^{\frac{-4}{2}} \implies (4)^{-2}\implies 4^{-2}\implies \cfrac{1}{4^2}\implies \cfrac{1}{16}$

Find the area of the composite figure:

Label: cm sq or centimeters squared

Answers

Answer:

I believe the awnser is 312 cm sq

Step-by-step explanation:

Did it on paper hope this helps

determine the equation of each radical function which has been tranformed

Answers

The equation of a radical function can be determined by using the transformation rules for x- and y- shifts, stretches/compressions, reflections, and rotations.

The equation of a radical function can be determined by applying various transformation rules. The most common transformations are shifts, stretches/ compressions, reflections, and rotations. For example, a radical function that is shifted left two units would have an equation of y = sqrt(x + 2). This is because the function is shifted two units to the left of the origin, so the equation must include an x + 2 term. Similarly, if the function is reflected in the x-axis, the equation would be y = -sqrt(x). This is because the reflection inverts the sign of the y-coordinate, so the equation must include a negative sign in the radical. For stretches and compressions, the equation must include a coefficient in front of the radical. For example, if the function is stretched horizontally by a factor of 3, then the equation would be y = 3 sqrt(x). This is because the function is stretched by a factor of three, so the equation must include a coefficient of 3 in front of the radical. Finally, for rotations, the equation must include a term for the angle of rotation. For example, if the function is rotated counterclockwise by 45 degrees, then the equation would be y = sqrt(x + y/tan45). This is because the angle of rotation is 45 degrees, so the equation must include a tan45 term.

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Solve the system by the method of your choice. Write your answer as a coordinate point (x,
y)
-6x + 5y = -27
- 3x + 3y = -115

Answers

(x,y) (-494/3, -203)

Please I need help!!
In the diagram DO and TO of rectangle OUTD are extended as shown.
Find the value of X
Show your work

Answers

Value of X is 8.

What is rectangle?

A rectangle is a closed two-dimensional figure with four sides. The opposite sides of a rectangle are equal and parallel to each other and all the angles of a rectangle are equal to 90°.

Given,

Figure OUTD is a rectangle.

Each angle in a rectangle measures 90 degrees.

Therefore,

(7X + 2) + 4X = 90

7X + 2 + 4X = 90

11X + 2 = 90

Subtract 2 from both sides of the equation

11X = 88

Divide both sides of the equation by 11

X = 8.

Hence, 8 is the value of x.

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Please I need help!! In the diagram DO and TO of rectangle OUTD are extended as shown. Find the value

What is equidistant from the three vertices of a triangle?

Answers

The three triangle vertices are equally spaced from the circumcenter of the triangle.

What is a cirumcentre?

All three vertices are intersected by the circle's centre. You can draw perpendicular bisectors of the triangle's sides and intersect them at the circumcenter to determine its location. The longest side of the triangle, sometimes referred to as the diameter, meets at the circumcenter. It is the centre of a circle that is tangent to each side of the triangle at its halfway, making it equally spaced from each vertex. In triangle geometry, the circumcenter is significant because it may be used to determine the circumradius, or circumcircle's radius.

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imagineaza ti o intamplare fantastica din timpul unei calatorii si redacteaza o naratiune de 100-150 de cuvinte in care sa prezinti. In compozitia ta.este necesar:-sa relatezi intamplarea,respectand succesiunea logica a evenimentelor; -sa prezici repele spati-temporale; -sa ai un continut adecvat cerintei; - sa respectti precizarea privind numarul de cuvinte

Answers

the answer is 50 maybe

I need help 9th grade math

Answers

Answer:

A. y - 2 = 2/3 (x - 0)

Step-by-step explanation:

x and y stand in a line at random with 10 other people. what is the probability that there are 3 people between x and y?

Answers

The probability that there are 3 people between x and y is 1/659470.

The total number of ways in which x and y can stand in a line with 10 other people is (12C2) = 66.

To find the number of ways in which there are exactly 3 people between x and y, we can treat x and y as a single entity and arrange them along with the 10 other people in 8 available spots in the line. This can be done in (8C1) = 8 ways.

Within this arrangement, there are 3 spots where x and y can be placed with 3 people between them,

Once x and y have been placed in one of these 3 spots, the remaining people can be arranged in the remaining spots in

10!/(10-8)! = 1814400 ways.

So the number of ways in which there are exactly 3 people between x and y is,

8 * 3 * 1814400 = 43545600.

Therefore, the probability that there are 3 people between x and y is,

43545600/66 = 659469.69 (rounded to 2 decimal places)

So, the probability is approximately 1 in 659470 or 1/659470.

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Find the unit vectors that are parallel to the tangent line to the parabola y = x2 at the point (6, 36).

Answers

The tangent line to the parabola y = x² at the point (6, 36) is (0.141, 1.68)

How to find the unit vector parallel to the tangent line

The equation of the tangent line at the point (6, 36) can be found using the derivative of the function and the point-slope form of a line:

f(x) = x²

f'(x) = 2x

y = f'(x) * (x - 6) + 36

Substituting f'(x) = 2x,

y = 2x * (x - 6) + 36

Now, we can find a unit vector that is parallel to this line by finding a non-zero multiple of the direction vector of the line, which is (1, 2x), and normalizing it.

To normalize, we divide by the magnitude, which is sqrt(1^2 + (2x)^2) = sqrt(1 + 4x^2).

Let's use x = 6:

y = 2 * 6 * (x - 6) + 36

y = 72

So, the direction vector is (1, 12) and its magnitude is sqrt(1 + 144) = sqrt(145) = 7.07.

The unit vector that is parallel to the tangent line is (1/7.07, 12/7.07) = (0.141, 1.68).

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(3x-17-4x^2)-(8x^2-5x+ 13)

Answers

Answer:$-12x^{2} + 8x - 30$

Step-by-step explanation:

Simplify the expression.

How do you graph y=3x-4 on a coordinate grid

Answers

Answer:

You can look at the screenshot

Step-by-step explanation:

So one point of the line is (0, -4) and another point is (4/3, 0). Plot these points on a graph and draw a line.

0, -4 is from making x = 0, and then 4/3, 0 is from making y = 0.

How do you graph y=3x-4 on a coordinate grid

Answer:

Step-by-step explanation:

You start the question at -4 on the coordinate plane and then you would go up 3 units and 1 unit to the right since the slope of the equation 3/1. You would get your graph like that. And to go backward you would go 3 units down and 1 unit to the left.

1. for x = 1001 1010 0011 1101, show the result of the following operations. a) shl(x) b) shr(x) c) cil(x) d) cir(x) e) ashl(x) f) ashr(x) g) dshl(x) h) dshr(x)

Answers

The following parts can be answered by the concept of Operations.

a) shl(x): The result of left shifting x by 1 bit is 0010 0100 0111 1010.

b) shr(x): The result of right shifting x by 1 bit is 0100 1100 0111 1011.

c) cil(x): The result of circularly left shifting x by 1 bit is 0010 0100 0111 1010.

d) cir(x): The result of circularly right shifting x by 1 bit is 1100 1101 1001 1110.

e) ashl(x): The result of arithmetic left shifting x by 1 bit is 0010 0100 0111 1010.

f) ashr(x): The result of arithmetic right shifting x by 1 bit is 1100 1101 1001 1110.

g) dshl(x): The result of double precision left shifting x by 1 bit is 0010 0100 0111 1010 0000.

h) dshr(x): The result of double precision right shifting x by 1 bit is 0000 1001 0100 1000 1111.

a) shl(x): Left shifting x by 1 bit means shifting all the bits in x to the left by 1 position. The leftmost bit is lost, and a 0 is shifted in from the right. Therefore, the result is 0010 0100 0111 1010.

b) shr(x): Right shifting x by 1 bit means shifting all the bits in x to the right by 1 position. The rightmost bit is lost, and a 0 is shifted in from the left. Therefore, the result is 0100 1100 0111 1011.

c) cil(x): Circularly left shifting x by 1 bit means shifting all the bits in x to the left by 1 position, and the leftmost bit is shifted to the rightmost position. Therefore, the result is 0010 0100 0111 1010.

d) cir(x): Circularly right shifting x by 1 bit means shifting all the bits in x to the right by 1 position, and the rightmost bit is shifted to the leftmost position. Therefore, the result is 1100 1101 1001 1110.

e) ashl(x): Arithmetic left shifting x by 1 bit is similar to logical left shifting, except that the sign bit (the leftmost bit) is preserved. Therefore, the result is 0010 0100 0111 1010.

f) ashr(x): Arithmetic right shifting x by 1 bit is similar to logical right shifting, except that the sign bit (the leftmost bit) is preserved. Therefore, the result is 1100 1101 1001 1110.

g) dshl(x): Double precision left shifting x by 1 bit means shifting all the bits in x, including the sign bit, to the left by 1 position. The leftmost bit is lost, and a 0 is shifted in from the right. Therefore, the result is 0010 0100 0111 1010 0000.

h) dshr(x): Double precision right shifting x by 1 bit means shifting all the bits in x, including the sign bit, to the right by 1 position. The rightmost bit is lost, and the sign bit is duplicated to fill the leftmost bit positions. Therefore, the result is 0000 1001 0100 1000 1111

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