The symmetry of a quadratic function is a vertical line that is x = − b/2a.
Quadratic function is defined as the function of the form f(x) = ax² + bx + c, where a, b and c are real numbers with a = 0 and it has the domain of a quadratic function is (−∞,∞).
Here we have given that we need to find the symmetry of a quadratic function.
As we all know that When a quadratic function is graphed in the coordinate plane, then the resulting parabola and corresponding axis of symmetry are vertical.
Here we have also know that in order to find the symmetry we have to replace y with (-y) and x with (-x) and the resulting function must be equal to the original function.
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How many solutions does x1 x2 x3 x4 x5 21 have?.
There are 11649 solutions for the given binomial series x¹ + x² + x³ + x⁴ + x⁵ = 21.
x¹ + x² + x³ + x⁴ + x⁵ = 21, where xi, i = 1, 2, 3, 4, 5, is a non-negative integer such that: 0≤ x1 ≤10 and x², x³, x⁴, x⁵ can range in between 0 to 21.
Now we have to choose the coefficient of x²¹ in the following expression
(1 + x + x² + x³.....+ x¹⁰) (1 + x + x² + x³.......+x²¹)(1 + x + x² + x³.....+ x²¹)(1 + x + x² + x³.....+ x²¹)(1 + x + x² + x³.....+ x²¹)
also we can extend term of x², x³, x⁴, x⁵ from x²¹ upto infinty because it didn't change the value of x²¹ coefficient .
(1 + x + x² + x³.....+ x¹⁰) (1 + x + x² + x³.......∞)(1+ x + x² + x³.....+ ∞)(1 + x + x² + x³.....+ ∞)(1 + x + x² + x³.....+ ∞)
(1 - x¹¹)/(1 - x)[1/(1 - x)⁴]
(1 - x¹¹) [1/(1 - x)⁵]
now we have to find only the coefficient of x²¹ for which we consider the coefficient of x²¹ and x¹⁰ in [tex]1-x^{-5}[/tex]
general term of the binomial coefficient is [tex]\binom{n+r-1}{r}[/tex]
[tex]\binom{5+21-1}{21}-\binom{5+10-1}{10}[/tex] = 12650-1001 = 11649
--The given question is incomplete; the complete question is
"How many solutions are there to the equation x¹ + x² + x³ + x⁴ + x⁵ = 21,
where xi, i = 1, 2, 3, 4, 5, is a nonnegative integer such that: 0≤ x¹≤ 10?"--
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The following rational equation has denominators that contain variables. for this equation,
A. write the value or values of the variable that makes the denominator zero. These are the restrictions on the variable. B. keeping the restrictions in mind, solve the equation.
3/x+4 + 2/x-4 = 16/(x+4)(x-4)
Answer:
A. x ≠ -4 or +4
B. no solution
Step-by-step explanation:
You want the restrictions on x and the solution to the equation ...
3/(x+4) +2/(x-4) = 16/((x +4)(x -4))
A. RestrictionsThe denominator factors are (x+4) and (x-4). The values of x that make these zero are -4 and +4, respectively.
The variable may not have values -4 or +4.
B. SolutionIn order to avoid extraneous solutions, it often works well to rewrite the equation in the form f(x) = 0. We can subtract 16/((x+4)(x-4)) from both sides to make that happen.
[tex]\dfrac{3}{x+4}+\dfrac{2}{x-4}-\dfrac{16}{(x+4)(x-4)}=0\\\\\\\dfrac{3(x-4)+2(x+4)-16}{(x+4)(x-4)}=0\\\\\\\dfrac{5x-20}{(x+4)(x-4)}=\dfrac{5(x-4)}{(x+4)(x-4)}=\dfrac{5}{x+4}=0[/tex]
There is no value of x that will make this true. The equation has no solution.
What is a polynomial equation explain and give 2 examples?.
A polynomial equation is an equation that consists two or more algebraic terms including at least one variable and constants through the algebraic operation ( addition, substraction, and multiplication).
A polynomial is defined as an equation that has at least one algebraic term with at least one variable and all exponents are integers greater than or equal to 0. The polynomial equation is expressed in the form of aₙ(x), here a is the coefficient, x is the variable and n is the exponent and expanded form is ; F(x) = aₙxₙ + aₙ₋₁ xₙ₋₁ + …….. + a₁x +a₀ = 0
Types of Polynomial Equation : A polynomial equation has mainally four types;
Monomial Equations : ax + b = 0Binomial Equations : ax² + bx + c = 0Cubic Equations : ay³ + by² + cy+ d= 0Quadratic Polynomial Equations : ax² + bx + c = 0 , a≠0 Cubic Polynomial EquationExample of polynomial equations are the following: 1) 3y² + 5y + 20 = 0
2) 5x² - 6x + 9 = 0
3) x² - 3x + 2 = 0
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Can the sides of a triangle have lengths 8, 16, and 18?
No, the sides of a triangle cannot have lengths 8, 16, and 18.
What is known as a triangle?
A triangle is a polygon with three vertices and three sides. The angles of the triangle are formed by the connection of the three sides end to end at a point. The triangle's three angles add up to 180 degrees in total.
This is because the length of any side of a triangle must be less than the sum of the other two sides and greater than the difference of the other two sides. This is called the triangle inequality theorem.
For a triangle with sides 8, 16, and 18, the side 8 is less than the sum of the other two sides 16+18=34, but greater than the difference of the other two sides 16-18=-2. Therefore, the sides of a triangle cannot have lengths 8, 16, and 18.
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What is the slope and y-intercept of the line whose equation is Y =- 2x 3?.
The provided line's slope and y-intercept are therefore 2, 3. We can use the details provided by the slope-intercept form to graph a linear equation.
For instance, the equation y=2x+3 reveals that the line's slope is 2 and that its y-intercept is located at (0,3).
How is the y-intercept determined?Finding the value of y at x=0 on a graph will reveal the y-intercept. The graph's intersection with the y-axis occurs at this location.
How can I determine the slope and y-intercept?Y = mx + b, where m denotes the slope and b the y-intercept, is how the equation of the line is expressed in the slope-intercept form. We can see that the slope of the line in our equation, y = 6x + 2, is 6.
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Full Question = What is the slope and y-intercept of the line whose equation is y=2x+3?.
Bernardo originally had 48 customers on his paper route. Through a newspaper sales promotion, his customer base increased to 56. What was the percentage of increase over the original number of customers?
Answer: 16.6667
Step-by-step explanation:
(48 × p) / 100 = 8
((48 × p) / 100) × 100 = 8 × 100
48p = 800
48p / 48 = 800 / 48
p = 16.6666666666667
Percent Increase ≈ 16.6667
What is the symbol of radical expression?.
Algebraic expressions involving radicals are known as radical expressions. The root of an algebraic expression makes up the radical expressions (number, variables, or combination of both).
The main symbol attached to any algebraic expression is x. Each radical expression consists of three separate components: first the "square root" sign itself serves as the radical sign.
The component of the expression inside the radical is known as the radicand
The degree outside the root indicates which root the radical refers to. Expressions without degree symbols have a degree of 2. A cube root is indicated by a degree of 3, a fourth root is shown by a degree of 4, and so on.
Now according to the above expression here, under the root can be any variable like, 2 , 28 , 25, or y7.
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What are examples of linear and nonlinear equations?.
Examples of linear equation are x + y = 7, 5x = 3 and nonlinear equations are x^2 - 8x + 3 = 0, x^7 + 5xy - 7 = 0.
The difference between linear and nonlinear equations lie in the degree of polynomial. If the degree of polynomial is 1, it is a linear equation and if the degree of polynomial is 2, it is a non linear equation.
There can be more than one variable in linear and non-linear equation.
Few examples of linear equations are
3x - 8 = 0
5y - 8x + 2 = 0
Few examples of non-linear equations are
3x² - 8 = 0
5y³ - 8x + 2 = 0
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The results of an experiment were listed in several numeric forms as listed below.
Order the numbers from least to greatest (least on top, greatest on bottom).
The order of the numbers from least to greatest is given by 0.003,1/5³,3/8,4/7,√5
What is a decimal number?A decimal number is a number that consists of a whole number and a fractional part separated by a point. For example – 3.5, 6.79, 78.32, etc.
Given here: The numbers are 1/5³ , 4/7,√5,3/8,0.003
To get a clarity of the magnitude of the numbers lets convert every number into decimal numbers.
Now we know To convert a fraction to a decimal, we'll just divide the numerator...by the denominator.
Thus 1/5³=1/125
=0.008
4/7=0.5714
3/8=0.375
√5=2.2361
and 0.003
Now arranging the numbers from in ascending order we get
0.003,0.008,0.375,0.5714,2.2361
Hence, The order of the numbers from least to greatest is given by 0.003,1/5³,3/8,4/7,√5
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Construct a square ABCD in which diagonal AC=6cm
Step-by-step explanation:
1. Draw the diagonal BD of length 6 cm as base.
2. Draw perpendicular bisector of DB which intersects DB at O.
3. Take O as centre and draw an arc of 3 cm above and below the line segment BD.
4. Join A, B, C and D.
Hence, ABCD is the requiredsquare.
see attachment....
Tipsy eats ½ ½ of a box a day
Cosmo eats ¼¼ of a box a day
Willow eats ⅛⅛ of a box a day.
If Mr. Dungan bought 6 boxes of cat food this week, will there be enough food to last the entire week?
Make a claim and explain to Mr. Dungan whether there will be enough cat food to last the entire week.
Reason:
1/2 = 4/8
1/4 = 2/8
1/2+1/4+1/8 = 4/8+2/8+1/8 = (4+2+1)/8 = 7/8
The three cats eat a combined 7/8 of a box per day.
Over the course of 7 days, they eat 7*(7/8) = 49/8 = 6.125 boxes of cat food. This value is slightly larger than 6, which means Mr. Dungan did not buy enough cat food. He'll need to buy a 7th box.
Side note: 6.125 = 6 & 1/8 as a mixed number.
An equilateral triangle has a perimeter of 12x+33. What is the length of each side of the triangle?
The length of each side of the triangle is 3 2/3 units
What is an equilateral triangle?An equilateral triangle can be defined as a type of triangle having all three sides of equal length.
It is also known as a regular triangle.
Unlike other triangles, an equilateral triangle is one having not only two sides equal like an isosceles triangle but all of its sides equal in length.
The formula for perimeter of an equilateral triangle is expressed as;
P = 3x
Where;
P is the perimeterx is the length of its sideSubstitute the values
12x + 33 = 3x
collect like terms
12x - 3x = 33
substract the like terms
9x = 33
Make 'x' the subject
x = 33/9
x = 3 2/3 units
Hence, the value is 3 2/3 units
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A small telephone company charges a $6.00 flat monthly fee, independent of the number of calls a customer makes. The telephone company does charge for the amount of time spent on the phone at the rate of 2.40 per hour. Fractional parts of an hour are billed accordingly. For example, one half hour costs $1.20 fifteen minutes cost $0.60 one minute costs $0.04 and so on Write and graph a function c(t) to model the monthly telephone cost as a function of the time t that the car is parked in the garage
explain and put the answer and
Question 2.A local parking garage charges $6.00 for up to and including 1 hour of parking. Each additional hour, or any part thereof, cost $2.40. Write and graph a function c(t) to model the cost of parking a car as a function of the time t that the car is parked in a garage
Can the sides of a triangle have lengths 5, 15, and 20?
Answer:
in what order are the sides in?
Answer:
No
Step-by-step explanation:
According to the triangle rule, the sum of the lengths of any two sides of a triangle is greater than the length of the remaining side. Let's check this condition:
5 + 15 > 20 ? -> No
So such a triangle cannot exist. And the sides can't be that lengths.
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Today, everytjing at a store is on sale. The store offers a 20% discount The regular price of a T-shirt is $18. What is the discount price? The discount price of a hat is $18. What is the regular price? If the regular price of an item is x dollars, what is the discount price in dollars
Answer:
1. The discount price of a T-shirt is $14.40
2. The regular price of a hat is $22.50
3. The discount price is y = x(0.8)
Step-by-step explanation:
1.
20% = 0.2
18 times 0.2 = $3.60
18 - 3.60 = $14.40
So, the discount price of a T-shirt is $14.40
2.
100% - 20% = 80%
18 / 80 = 0.225
0.225 x 100 = $22.50
So, the regular price of a hat is $22.50
3.
Let's y represent the total cost
We have the equation
y = x(0.8)
What is the 5th equivalent fraction to 1 5?.
The 5th equivalent fraction to 1/5 is 5/25.
An equivalent fraction is a fraction that has the same value as another fraction, but a different numerator and denominator. There are an infinite number of equivalent fractions for any given fraction. To find the 5th equivalent fraction to 1/5, you can use the fact that equivalent fractions have a common multiple as a denominator.
The first step is to find a common multiple of the denominators of the fractions you want to compare. In this case, the denominator of the first fraction is 5, and you are looking for the 5th multiple of 5, which is 5*5 = 25.
The next step is to find the numerator that corresponds to the common denominator found in step 1. To do this, you can use the concept of cross-multiplication: the product of the original denominator and the new numerator is equal to the product of the original numerator and the new denominator.
1/5 * 5/5 = 1 * 5/5 * 5 = 5/25
Therefore, the 5th equivalent fraction to 1/5 is 5/25.
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What do you need to prove in proof by contradiction method?.
Proof by contradiction is a kind of proof that demonstrates the truth or the validity of a proposition by showing that assuming the proposition to be false leads to a contradiction.
More broadly, proof by contradiction is a form of argument that shows a statement by arriving at a contradiction, even when the initial assumption is not the disproof of the statement to be proved. In general, proof by contradiction is also known as indirect proof.
A mathematical proof utilising proof by contradiction usually proceeds as follows:
1. The proposition to be proved is P.
2. We consider P to be false, i.e., we assume ¬P.
3. It is then shown that ¬P implies falsehood. This is typically accomplished by deriving two mutually contradictory assertions, Q and ¬Q, and appealing to the Law of noncontradiction.
4. Since assuming P to be false leads to a contradiction, it is concluded that P is, in fact, true.
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Rectangle A B C D is transformed by a combination of two transformations so that all points on AB are invariant and there are no other invariant points.
The first transformation is
a reflection in the line with equation y = k, where ‘k’
is an integer and k ≠1
Describe fully the second transformation.
(2 marks)
It is a translation by vector
(?)
Dilation is the process by which rectangle ABCD becomes another rectangle ABCD.
What is invariant transformations?When the predictions of physical laws remain unaltered, they are considered to be invariant under transformations.This typically means that no new or different solutions are discovered since the law's form (such as the kind of differential equations used to represent it) remains unchanged via modifications.An altered rectangle's dimensions and orientation are called transformation.Dilation is the process by which rectangle ABCD becomes another rectangle ABCD.We infer from the question that the sizes of the two figures are proportionate rather than equal.The transition is a dilation since the sizes are proportionate but not equal.Consequently, (a) dilation is the best choice.To learn more about invariant transformations refer
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the point (4, -3) is reflected across the y-axis. graph the new coordinate.
-4,-3
Step-by-step explanation:
If you reflect over the y axis only the x #
Ann and ben translate documents from german into english. A set of documents that would take ann 8 days would take ben 12 days. Ann starts to translate documents. After 2 days ann and Ben both work on translating the documents. How many more days will it take to complete the work? You must show your working
The time that it will take for them to complete the work is given as follows:
2 and 17/20 days.
How to obtain the time it will take for them to complete the work?To obtain the remaining time, we use the together rate, which is the sum of each of the separate rates.
The rates for this problem are given as follows:
Ann: 1/8.Ben: 1/12.Together: 1/x.Hence the together rate is obtained as follows:
1/x = 1/8 + 1/12
1/x = 5/24
5x = 24
x = 3.8 days.
Ann worked two days, meaning that only 6/8 = 75% of the document has to be completed, hence the time is of:
0.75 x 3.8 = 2.85 days = 2 and 17/20 days.
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is 5/8 less then 1/2
Answer:no
Step-by-step explanation:
Let's start with the first method: converting the fractions to the same denominator. First, we'll set up 5/8 and 1/2 side by side so they are easier to see:
Converting Denominator
5
8
&
1
2
Our denominators are 8 and 2. What we need to do is find the lowest common denominator of the two numbers. This is the smallest number that can be divided by both 8 and 2. In this case, the lowest common denominator is 8.
If we multiply the first denominator (8) by 1 we will get 8. If we multiply the second denominator (2) by 4 we will also get 8. We also need to multiply the numerators above the line by the same amounts so that the fraction values are correct:
5 x 1
8 x 1
1 x 4
2 x 4
This is what 5/8 and 1/2 look like with the same denominator:
5
8
&
4
8
Now that these fractions have been converted to have the same denominator, we can clearly see by looking at the numerators that 5 is NOT less than 4 which also means that 5/8 is NOT less than 1/2.
zach read 11/20 of his assignment. what percent of his assignment did he read? PLEASE HELP WILL GIVE BRAINLIST IF SOLUTIONS
Answer:
55%
Step-by-step explanation:
11/20=0.55
0.55 is the same as 55% since 55/100 is 0,55.
george places a mirror on the ground 45ft from the base of building. He walks backward until he can see the top of the building in the middle of the mirror. At that point, George’s eyes are 6th above the ground and he is 7.5ft from the mirror. Use similar triangles to find the height of the building.
The height of the building is 36 ft.
What are similar triangles?Similar triangles have the same corresponding angle measures and proportional side lengths.
Given that, George places a mirror on the ground 45ft from the base of the building. He walks backward until he can see the top of the building in the middle of the mirror. At that point, George’s eyes are 6th above the ground, and he is 7.5ft from the mirror.
As we can see, this arrangement is creating similar triangles,
We know that, similar triangles have corresponding sides in equal ratios,
Therefore,
x / 45 = 6 / 7.5
x = 6 / 7.5 × 45
x = 36
Hence, the height of the building is 36 ft.
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what is the
number if you divide 10 and = 30
Answer: 300
Step-by-step explanation: if ur dividing and you don’t know the divisible, you multiply the answer by the number you divide with, so 10(30) equals 300.
Determine whether the side
lengths form a triangle
3 cm, 8 cm, 12 cm
The side with length 5 cm is the hypotenuse.
The given lengths of the sides are 3 cm,4 cm and 5 cm
3²=3×3=9;
4² =4×4=16;
5² =5×5=25;
We found that
9+16=25
3²+4²+5²
Therefore, the triangle is a right-angled triangle.
Note : In any right-angled, triangle the hypotenuse happens to be the longest side.
In this example the side with length 5 cm is the hypotenuse.
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the bases of a prism are right triangles with side lengths 12 meters, 15 meters, and 19 meters. The total surface area of the triangle prism is 1192 square meters. What is the height of the prism of the triangle prism
The height of the triangular prism is 22 meters, having a total surface area of 1192 square meters.
What is an equation?An equation consists of numbers and variables linked together by mathematical operations to form an expression.
Let h represent the height of the triangular prism.
The base are right triangles with side lengths 12 meters, 15 meters, and 19 meters. Hence:
Total surface area = 2 * Area of base + sum of remaining area of surface
Total surface area = 2(1/2 * 15 * 12) + 12*h + 15*h + 19*h
The total surface area of the triangle prism is 1192 square meters. therefore:
1192 = 180 + 46h
46h = 1012
h = 22
The height is 22 meters
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In a certain town, 62% of the residents enjoy riding roller coasters. Alfred takes a sample of 75 residents from this
town. What is the probability that fewer than 58% of the residents in the sample enjoy riding roller coasters?
Find the z-table here.
O 0. 238
O 0. 433
O 0. 714
O 0. 762
The probability that fewer than 58% of the residents in the sample enjoy riding roller coasters is b) 0.433.
What is probability ?
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true.The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty.
The mean value is 0.62
Z value formula is
z = (x' -mean)/([tex]\sqrt{n}[/tex])
Substituting the given values, we get -
sigma = Sqrt {[tex](x'-mean)^2[/tex]/n}
Sigma = sqrt { [tex](0.58-0.62)^2[/tex]/75} = 0.09
z = (0.58-0.62)/(0.09)
z = -0.433
Hence , The probability that fewer than 58% of the residents in the sample enjoy riding roller coasters is b) 0.433.
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The likelihood that less than 58% of the sample's residents enjoy going on roller coasters is 0.433.
Probability is the possibility that something will occur. The total number of conceivable outcomes.
The probability of an event is a number between 0 and 1, where, broadly speaking, 0 represents the event's impossibility and 1 implies certainty.
The average is 0.62.
The formula for the Z value is
Z = (x' – mean) ([tex]\sqrt{n}[/tex])
By substituting the provided values, we get -
Sigma Equals [tex]\sqrt{\frac{(x' - mean)^2}{n} }[/tex]
Sigma is equal to sqrt {[tex]{(0.58-062)^2/75[/tex]} = 0.09
z = (0.58-0.62)/(0.09) (0.09)
z = -0.433
Therefore, the likelihood that less than 58% of the sample's residents enjoy riding roller coasters is 0.433.
Option B is the proper response, thus.
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What are the types of complex numbers?.
Any complex number can be divided into four categories based on the characteristics of the real component and imaginary part:
imaginary number.complicated number 0completely fictitious numberall-real numbersComplex numbers are those that are expressed as a+ib, where a and b are actual numbers and I is a fictitious number called a "iota." I has the value (-1). As an illustration, the complex number 2+3i is made up of the real number 2 (Re) and the imaginary number 3i (Im).
The real and imaginary halves of a complex number are represented by the first and second values, respectively, of the complex data type in Python.
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cuantas veces cabe 12 en 77?
Answer:
6.4166
Step-by-step explanation:
77/12 = 6.41666666667
What is the first step in simplifying the logarithmic expression log2 (x-3) + log4x?
[tex]\textit{Logarithm Change of Base Rule} \\\\ \log_a b\implies \cfrac{\log_c b}{\log_c a}\qquad \qquad c= \begin{array}{llll} \textit{common base for }\\ \textit{numerator and}\\ denominator \end{array} \\\\\\\begin{array}{llll} \textit{logarithm of factors} \\\\ \log_a(xy)\implies \log_a(x)+\log_a(y) \end{array} ~\hspace{4em} \begin{array}{llll} \textit{Logarithm of exponentials} \\\\ \log_a\left( x^b \right)\implies b\cdot \log_a(x) \end{array} \\\\[-0.35em] \rule{34em}{0.25pt}[/tex]
[tex]\log_2(x-3)+\log_4(x) \\\\[-0.35em] ~\dotfill\\\\ \log_4(x)\implies \stackrel{\textit{change of base rule}}{\cfrac{\log_2(x)}{\log_2(4)}}\implies \cfrac{\log_2(x)}{\log_2(2^2)}\implies \cfrac{\log_2(x)}{2}\implies \cfrac{1}{2}\log_2(x) \\\\[-0.35em] ~\dotfill\\\\ \log_2(x-3)+\log_4(x)\implies \log_2(x-3)+\cfrac{1}{2}\log_2(x)\implies \log_2(x-3)+\log_2(x^{\frac{1}{2}}) \\\\\\ \log_2(x-3)+\log(\sqrt{x})\implies {\Large \begin{array}{llll} \log_2( ~~ \sqrt{x}(x-3) ~~ ) \end{array}}[/tex]
