Write each equation in logarithmic form.
10⁴=10,000

The solutions to the proportional equation are given as follows:

x = -4 and x = 5.

A proportion is a fraction of a total amount, and equations can be built to find the desired measures in the context of the problem using operations such as multiplication or division.

One example of application of proportions is for the solution of equations when two sides are proportional, meaning that multiplication operations are applied between the opposite sides to find the unknown variable.

For this problem, we already are given the following proportional equation:

3x/5 = 12/(x + 1).

Due to the equality sign, we can apply cross multiplication, hence:

3x(x + 1) = 60

3x² + 3x = 60

3x² - 3x - 60 = 0

x² - x - 20 = 0

(x - 5)(x + 4) = 0.

Hence the solutions are found as follows:

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Answer:

I think it's $15 I hope this helps

Answer:

$15

Step-by-step explanation:

If 14 cards cost $6, then 7 cards will cost $3

So 35 cards will cost: $3 x 5 = $15

The domain of the function is (2,∞),{x|x>2}

and the range of function is  (−∞,∞),{y|y∈R}

The domain is all values of xx that make the expression defined.

Set the argument in log(x−2)greater than 0 to find where the expression is defined.x−2>0

Add 2 to both sides of the inequality.x>2

Interval Notation:(2,∞)

Set-Builder Notation:{x|x>2}{x|x>2}

The range is the set of all valid y values.

We can find out the range by using the graph.

Interval Notation:(−∞,∞)

Set-Builder Notation:{y|y∈R}

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According to the question, the given rational expression can be simplified and as shown below:

Rational expression = [tex]\frac{x^{2} -7x+10}{x^{2} -8x+15}/\frac{4-x^{2} }{x^{2} +3x-18}[/tex]

To simplify the given rational expression, calculate the factors of the numerator as well as denominator. And later divide it from each other. These expressions are also called as polynomial expressions.

Rational expression = [tex]\frac{(x-5)(x-2)}{(x-3)(x-5)}/\frac{(2-x)(2+x)}{(x+6)(x-3)}=\frac{-(x+2)}{(x+6)}[/tex]

The final expression of the given equation is [tex]\frac{-(x+2)}{(x+6)}[/tex] .

What is polynomial expression?

Polynomial expression are those equation whose highest degree is two. It is also called as quadratic equation. And it is simplified by performing the factors on the given expressions.

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Answer: 2.2

Step-by-step explanation: twenty percent of 11 is 2.2

Point A' after the dilation of the scale factor by 1/3 is (-1,2) is the correct option.

The given point is;

Point A = (3, -6), scale factor = 1/3

Fortunately, because the dilation is centered at the origin (0,0), we can multiply the scale factor by the x and y-coordinates to acquire the image point coordinates.

Point A' = (3×1/3, -6×1/3)

Point A' = (1, - 2)

As a result, if it grows larger, it should start moving away from the origin, while if it grows smaller (as it does here), this should move closer to a origin.

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Each logarithm as the quotient of two common logarithms is :

[tex]\frac{log\\ 3x}{log \\4}[/tex]

In, mathematics a logarithmic equation is inverse of exponential equation. That means, we can easily convert the logarithmic equation into exponential equation and vice versa. The basic form of the logarithm function can be written as  .

   [tex]log_ax=N[/tex]

where, a is the base of function or equation

           x define expression

Mathematically, if  [tex]a^N=x[/tex]  is an exponential function then, this can be converted to logarithmic function as  [tex]log_ax=N[/tex].

There are many formulas that are given to us for solving several simple and complex problems based on logarithmic functions and logarithmic equations.

Some important formulas of logarithms are :

The given expression is,

log₄3 x

as, [tex]log_bm= \frac{log_cm}{log_cb}[/tex]

therefore ,

log₄3 x =  [tex]\frac{log \ 3x}{log \ 4}[/tex]

Hence,  logarithm as the quotient of two common logarithms is :

[tex]\frac{log\\ 3x}{log \\4}[/tex]

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The exponent property that should be applied is the quotient of a power property. Option A

Index form of a number can be defined as the power or exponent raised to a number or variable.

It is also known as the standard form of a number.

The quotient of a power property states that the power of a quotient is equivalent to the quotient that is obtained when the numerator and denominator are each raised to a particular power differently, before the division is carried out.

It is used to simplify the problem of division involving exponents.

Given the expression;

3r³b^5/9b

Take the inverse exponent of the divisor

3r³b^5 × b^-1/ 9

Divide the common factors

r³b^5 × b^-1/3

Add the common exponents

r³b^4/3

Thus, the exponent property that should be applied is the quotient of a power property. Option A

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Answer:

180

Step-by-step explanation:

Making t the subject of formula gives the following expression: t = (2 - 3q)/(q + 4).

An expression can be defined as a mathematical equation which is used to show the relationship existing between two or more variables and numerical quantities.

In this exercise, you're required to describe the steps that should be taken to make t the subject of formula. Therefore, the appropriate and required steps are listed as follows:

q = (2 - 4t)/(t + 3)

Cross-multiplying, we have:

q(t + 3) = (2 - 4t)

qt + 3q = 2 - 4t

Collecting like terms, we have:

qt + 4t = 2 - 3q

Factorizing t, we have:

t(q + 4) = 2 - 3q

Dividing by (q + 4), we have:

t = (2 - 3q)/(q + 4)

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The missing value of direct variation is 10

The direct variation property states that , whenever the division of values of two quantities is a constant value then there is same variation between these two corresponding values. Or we can also say that if the divisible nature of two corresponding values is constant, then one quantity is directly varies with second quantity. That mean if one quantity increase other will also increase and vice versa.

Let say if x and y are two values and k is a constant value, then

According to direct variation

k = y/x or y = k x

In the given question,

Take the order pair (2,5)

As it is an direct variation, we can substitute it

y=k x  

5=k (2)

K=2.5

Now find the direct variation equation

y=k x

Also, k=2.5

Solve this equation for the second pair (4,y)

As, y=2.5 x

   y= 2.5 (4) = 10

Therefore, missing value is 10

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I found the graph, the answer should be 25$

Answer:

32

Step-by-step explanation:

Since 1 cup is equal to 4 pancakes would need to multiply 8 cups by 4 pancakes (how many you make with one cup) Which will give you 32. You can check by dividing 32 by 8 or 4 and you will get one of the numbers.

Hope that helps, please comment if you need anything else :D

Using a linear function, it is found that the least possible number of matches played when four players are left is of 339.

A linear function is modeled according to the following rule:

y = mx + b

In which:

For the situation modeled in this problem, we have that:

Thus the average number of players after n matches is given by:

P(n) = 117 - 1/3m.

The least number of matches for there to be 4 players is found when P(n) = 4, hence:

4 = 117 - 1/3m.

1/3m = 113.

m = 113 x 3

m = 339.

Hence at least 339 matches must be played.

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Answer:

Huh?

Step-by-step explanation:

1,2,3,4,5,6,7,8,9,11,12,13,14,15,16,17,18,19 (?)

ln 1 is the converted from the expression 2 ln 8-3 ln 4 to a single natural logarithm.

A number's natural logarithm is its logarithm to the base of the transcendental and irrational number e, which is roughly equivalent to 2.718281828459.

Typically, the natural logarithm of x is denoted by the symbols ln x, loge x, or, if the base e is implicit, just log x. When adding parentheses for clarity, the values ln(x), loge(x), or log (x). In order to avoid ambiguity, this is notably done when the argument to the logarithm is not a single symbol.

The power to which e would need to be raised in order to equal x is the natural logarithm of x.

For instance, e2.0149... = 7.5, thus ln 7.5 equals 2.0149.

Since e1 = e, the natural logarithm of e itself, or ln e, is equal to 1, while the natural logarithm of 1 is 0, since e0 = 1.

We need to convert 2 ln 8-3 ln 4 as a single natural logarithm

First apply power property

⇒ ln 8² - ln 4³

⇒ ln 64 - ln 64

Now, applying quotient property

⇒ ln [tex]\frac{64}{64}[/tex] = ln 1

Here, we finally converted 2 ln 8-3 ln 4 to a single natural logarithm ln 1

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Answer:

C

Step-by-step explanation:

lets start with the first equation

33*33^7

Since they are not in parentheses, the only one going the power of 7 is the second 33. The first 33 will just have n exponent of one, so you add those together and get 33^8

The next equation is explained like this

(33*33)^7

Since this equation is in parentheses, first you will combine the 33:

(33^2)^7

Then, you will follow power to a power rule, and multiply the 2 and 7 to get this answer:

33^14

Then, find the awnser that matches.

Answer: -3,6 to graph it you would mark it at -3 intall you reach to 6

Step-by-step explanation: hope this helps

Answer:

Step-by-step explanation:

-21<3x-12            3x-12<6

-12      -12                +12 +12

-33<3x                3x<18

divide 3              divide 3 both sides

both sides          x<6

-11<x

swap the < to >

x>-11

Answer:

[tex] \sf \fbox{Age of pamela is 8 year}[/tex]

Step-by-step explanation:

let the age of Debra is x and Pamela is x+7 year old,

Given condition- In 6 years Pamela will be twice as old as debra, so the algebraic expression will look like,

[tex]x+7+6 = 2(x+6)[/tex]

[tex]x+13 = 2x + 12[/tex]

[tex]2x-x = 13-12[/tex]

[tex]x = 1[/tex]

Age of debra is 1 years,

now we can calculate the age of pamela by substituting value of x,

x + 7 = 1 + 7 = 8 years

━━━━━━━━━━━━━━━━━━━━━━━

Verification-

after 6 years the age of debra & pamela will be 7 & 14. 14 is twice of 7 which satisfies the given condition.

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In the given set, 9, 16, 25, 49, 64, 100 are perfect squares.

What is perfect square?

A square number, sometimes known as a perfect square, is an integer that is the square of another integer. It is the result of multiplying another integer by itself.

For the number 9 = 3 [tex]\times[/tex] 3 = 3^2

9 is a perfect square number.

For the number 12 = 3 [tex]\times[/tex] 4

For the number 16 = 4 [tex]\times[/tex] 4 = 4^2

16 is a perfect square number.

For the number 25 = 5 [tex]\times[/tex] 5 = 5^2

25 is a perfect square number.

For the number 32 = 2 [tex]\times[/tex] 2 [tex]\times[/tex] 2 [tex]\times[/tex] 2 [tex]\times[/tex] 2 = 2^5

For the number 49 = 7 [tex]\times[/tex] 7 = 7^2

49 is a perfect square number.

For the number 56 = 7 [tex]\times[/tex] 8 = 7 [tex]\times[/tex] 2^3

For the number 64 = 8 [tex]\times[/tex] 8 = 8^2

64 is a perfect square number.

For the number 100 = 10 [tex]\times[/tex] 10 = 10^2

100 is a perfect square number.

Therefore, in the given set, 9, 16, 25, 49, 64, 100 are perfect squares.

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According to  the given equation (e⁰=1) the logarithmic form  log[tex]_e[/tex](1) = 0.

The logarithmic mean is indeed a component of multiple non-negative values that is equal to the respective difference divided by their quotient's logarithm. This computation is useful in heat and mass transfer problems in engineering.

Logarithms depict changes in terms or multiplication: each step in the examples above is 10x larger. Each step in the natural log is "e" (2.71828...) times longer. When dealing with a succession of multiplications, logarithms assist in "counting" them, just as addition does when effects are added.

e⁰=1

= log[tex]_e[/tex]⁰1

the given equation (e⁰=1) the logarithmic form  log[tex]_e[/tex](1) = 0.

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Yes , the given function is an inverse function which is given by  [tex]f^{-1}(y)=\frac{y-2}{\sqrt{2} }[/tex] .

We have given a function [tex]f(x)=x\sqrt{2} -1+3[/tex] .        ....(1)

On simplifying the equation we, get

     [tex]f(x)=x\sqrt{2} +2[/tex]

Let y = f(x)

      [tex]y=x\sqrt{2} +2\\\\y-2=x\sqrt{2}\\\\x=\frac{y-2}{\sqrt{2} }[/tex]           ......(2)

Swap the values of x and y, then the inverse is then

     [tex]x=f^{-1}(y)[/tex]     ...(3)

Comparing equation (2) and (3) , we get

  [tex]f^{-1}(y)=\frac{y-2}{\sqrt{2} }[/tex]

We can check that this is really an inverse function or not

Putting [tex]x=f^{-1}[/tex] in equation (1) , we get

   [tex]f(f^{-1}(y))=\frac{y-2}{\sqrt{2} } \sqrt{2} -1+3\\\\y=y-2+2\\\\y=y[/tex]

 As LHS = RHS  which means inverse function is correct .

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Answer:

the answer is the 10⁰ ok that

The simple way to solve such an expression; −(−8q−8.7) as given in the task content is by the distributive property which yields; 8q + 8.7.

It follows from the task content that the expression which is to be simplified, or perhaps written in a different form is; −(−8q−8.7).

Therefore, it follows from the distributive property that we have;

-(-8q) -(-8.7)

Also, since the product of two negative signs yields a positive sign.

Hence, we have; 8q + 8.7 as the equivalent of the expression given.

Ultimately, the simple way to solve such an expression; −(−8q−8.7) as given in the task content is by the distributive property which yields; 8q + 8.7.

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Using the midpoint formula, the coordinates of the midpoint M is: (-1/2, 1).

Using the distance formula, the length of YZ is: 14.3.

The midpoint formula is: M[(x1 + x2)/2, (y1 + y2)/2].

Distant formula is: d = [tex]\sqrt{(y_2 - y_1)^2 + (x_2 - x_1)^2}[/tex].

Given the following:

Y(1,-6) = (x1, y1)

Z(-2,8) = (x2, y2)

Apply the midpoint formula to find the midpoint of YZ:

M(x, y) = [(1 + (-2))/2, (-6 + 8)/2].

M(x, y) = (-1/2, 2/2)

M(x, y) = (-1/2, 1)

M(-1/2, 1)

Apply the distance formula to find YZ:

YZ = √[(−2−1)² + (8−(−6))²]

YZ = √[(−3)² + (14)²]

YZ = √205

YZ ≈ 14.3 (nearest tenth)

In conclusion:

Using the midpoint formula, the coordinates of the midpoint M is: (-1/2, 1).

Using the distance formula, the length of YZ is: 14.3.

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The Range of the given function weight is; [4,000 , 5,800]

The range of a function is defined as the set of all possible output values of that function.

We are Given:

Function; W = 6F + 4000

Full amount of fuel = 300 gallons

No amount of fuel = 0 gallons

Thus, the set of all possible input values which is the domain is;

Domain : amount of fuel = [0 , 300]

If F = 0, then we have;

W = 6F + 4,000

W = 6(0) + 4,000

W = 4,000

If F = 300, then we have;

W = 6F + 4,000

W = 6(300) + 4,000

W = 5,800

Thus, we conclude that the Range of weight is; [4,000 , 5,800]

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Complete question is;

The fuel tank of a certain airplane can hold up to 300 gallons of fuel. Let Wbe the total weight(in pounds) Let F be the total amount of fuel in this tank ( in gallons) Suppose that W = 6F +4000 gives W as a function of F

The answer to the question is c 47,625

The concept used is geometric progression.

A geometric progression is a sequence in which any element after the first is obtained by multiplying the preceding element by a constant called the common ratio which is denoted by r.

Here we can see that shares are being halved per week

we can say that the value of r is 1/2

so we can use the formula of sum of geometric progression to calculate the total number of shares.

sum of geometric progression formula =a(1-r∧n)/1-r

Here we can see that the value of n is 7

On putting the values in the formula we get

sum=24,000(1-1/128)/(1/2)

On further solving we get-

24000×127×2/128 which is equal to 47,625

Hence the total no of shares  of the company he bought are 47,625

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The answer is  102.93174866

The Karatsuba set of rules becomes the primary multiplication set of rules asymptotically quicker than the quadratic "grade faculty" algorithm. The Toom–cook algorithm (1963) is a faster generalization of Karatsuba's technique, and the Schönhage–Strassen algorithm (1971) is even quicker, for sufficiently big n.

Multiplication is the main device for lots of forms of maths which include algebra, calculus, equations, and greater. The capacity to rehearse and understand multiplications up to and such as 12 by means of the final yr of number one college will permit your toddler to with a bit of luck and skillfully address greater complex mathematical topics.

Familiarity and skillability with the primary timetables are essential building blocks in math. It opens the door to multi-digit multiplication and demystifies tactics like long department and simplifying fractions. It lays the foundation for algebra.

Multiply 10 by

10.100+10(−√3)+√6⋅10+√6(-√3)

Multiply −1 by 10.100-10√3+√6⋅10+√6(−√3)

Move 10 to the left of

√6.100-10√3+10.√6+√6(−√3)

Multiply √6(−√3).

Combine using the product rule for radicals.

100−10[tex]\sqrt{3}[/tex]+10[tex]\sqrt{6}[/tex]−[tex]\sqrt{6.3}[/tex]

Multiply 6 by

3.100−10[tex]\sqrt{3}[/tex]+10[tex]\sqrt{6}[/tex]−[tex]\sqrt{18}[/tex]

Rewrite 18 as 32⋅2.

Factor 9 out of

18.100−10[tex]\sqrt{3}[/tex]+10[tex]\sqrt{6}[/tex]−[tex]\sqrt{9}[/tex](2)

Rewrite 9  as

32.100−10[tex]\sqrt{3}[/tex]+10[tex]\sqrt{6}[/tex]−[tex]\sqrt{32.2}[/tex]

Pull terms out from under the radical.

100−10[tex]\sqrt{3}[/tex]+10[tex]\sqrt{6}[/tex]−(3[tex]\sqrt{2}[/tex])

Multiply 3 by −1.100−10[tex]\sqrt{3}[/tex]10[tex]\sqrt{6\\}[/tex]−3[tex]\sqrt{2}[/tex]

The result can be shown in multiple forms.

Exact Form:

100−10[tex]\sqrt{3}[/tex]+10[tex]\sqrt{6\\}[/tex]−3[tex]\sqrt{2}[/tex]

Decimal Form:

102.93174866

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The simplified form of 1 and 3 over 4 times, 2 and 2 over 3 is 4/3.

Given,

1 and 3 over 4 times

2 and 2 over 3 times

We can convert this to mathematical form:

That is,

1 and 3 over 4 times = (1 + 3) / 4

2 and 2 over 3 times = (2 + 2) / 3

Now, equate the above terms:

We get,

(1 + 3) / 4 x (2 + 2) / 3

= (4 / 4) x (4 / 3)

= 1 x (4 / 3)

= 4 / 3

That is, the simplified form of 1 and 3 over 4 times and 2 and 2 over 3 times is 4/3.

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