Use the product, quotient, and power rules of logarithms to rewrite the expression as a single logarithm. Assume that all variables represent positive real numbers. 4log x + 2log y

The vertical asymptotes are x = 4 and x = 2, and the horizontal asymptote is y = -3.

The given function is f(x) = −3(x + 3)(x − 2) / (x − 4)(x − 2).

To find the vertical asymptotes, we need to determine the values of x for which the function approaches positive or negative infinity. The denominator of the rational expression cannot be equal to zero, so we set it equal to zero and solve for x: (x − 4)(x − 2) = 0. This gives us two vertical asymptotes at x = 4 and x = 2.

To find the horizontal asymptote, we need to examine the highest power of x in the numerator and denominator. In this case, both the numerator and denominator have the same highest power, which is x^1 or simply x. Therefore, to find the horizontal asymptote, we compare the coefficients of the highest power of x. The coefficient of x in both the numerator and denominator is -3. So the horizontal asymptote is y = -3.

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The vertical asymptotes of this rational function are x = 4 and x = 2.

The horizontal asymptote is y = -3.

In Mathematics and Geometry, a rational function refers to a type of function which is expressed as a fraction.

By critically observing the rational function [tex]f(x)=\frac{-3(x + 3)(x - 2)}{(x -4)(x - 2)}[/tex] shown above, we can logically deduce that its vertical asymptote can be determined by solving the denominator as follows;

(x - 2)(x - 4) = 0

x = 2 and x = 4.

Since the degree of the numerator is the same as the degree of the denominator, we can reasonably infer and logically deduce that this rational function's horizontal asymptote can be calculated by dividing the leading coefficient of the numerator by the leading coefficient of the denominator as follows;

y = -3/1

y = -3.

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Answer:

=

Step-by-step explanation:

When solving this, we just have to remember that the zero´s to the right after the decimal point are meaningless, so basically 0.700 is the same as 0.7 so in this case both number are the same, 0.700 and 0.7 are the same number expressed in different ways.

Final answer:

The expression –32 + (2 – 6)(10) evaluates to –72, by first calculating the operation within the parentheses, multiplying it by 10, and then adding it to –32.

Explanation:

To evaluate the expression –32 + (2 – 6)(10), we must follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Firstly, we calculate the value inside the parentheses (2 – 6), which equals –4. Then we multiply this value by 10 to get –40. Finally, we add the result to –32 which gives us –32 + (–40) = –32 – 40 = –72.

Final answer:

The expression –32 + (2 – 6)(10) evaluates to –372 after performing the operations inside the parentheses and then multiplying and adding as per the order of operations.

Explanation:

To evaluate the expression –32 + (2 – 6)(10), we start by performing the operation inside the parentheses first. We subtract 6 from 2, getting –34. Then we multiply this result by 10 to get –340.

Next, we add this result to –32: –32 + (–340) = –372. So the final answer to the expression is –372.

Option C correctly applies the Law of Syllogism. By combining the hypothesis from the first statement, 'If it is springtime, you will plant flowers in the garden' with the conclusion of the second, 'If you plant flowers in the garden, your yard will smell good' we form 'If it is springtime, then your yard will smell good'.

The question deals with logical reasoning, specifically the laws of syllogism and detachment. The Law of Syllogism takes two conditional statements and forms a conclusion by combining the hypothesis of one statement with the conclusion of another. The Law of Detachment, also known as Modus Ponens, applies when a single conditional statement and the affirmation of its hypothesis lead to the affirmation of its conclusion.

Looking at the options:
A. The statement does not apply the law of syllogism.
B. The statement does not apply the law of detachment.
C. This statement correctly applies the Law of Syllogism. We have two statements: 'If it is springtime, you will plant flowers in the garden' and 'If you plant flowers in the garden, your yard will smell good'. Combining the hypothesis of the first with the conclusion of the second gives us 'If it is springtime, then your yard will smell good'.
D. This accurately applies the law, but the law it uses is not mentioned.

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Polaris is the basis of our calculation having a deviation of 0.7 degrees from the fixed north. So if the angle between Polaris and the horizon is 40 degrees, then it only means that you are at approximately 39.3-40.7 degrees north latitude.

Final answer:

The angle between Polaris and the horizon is approximately equal to the observer's latitude. Therefore, if Polaris is observed at a 40° angle from the horizon, the observer is at approximately 40° North latitude.

Explanation:

When you observe Polaris, the North Star, from your location and find that its angle above the horizon is 40°, this angle is equivalent to your approximate latitude. This is because Polaris is situated nearly directly above the Earth's North Pole.

Therefore, the altitude angle of Polaris, as you observe it from a particular location in the Northern Hemisphere, generally corresponds to that location's latitude. In your case, since Polaris is at a 40° altitude, your latitude is approximately 40° North.

IQ scores follow a bell curve distribution where 100 is the mean. About 50% of people score above 100. Therefore, in a group of 78 students, we would expect about 39 students to have IQ scores above 100.

The question is asking to determine what percent of 78 students in a population have IQ scores above a 100. Given that the IQ scores in large populations are centered at 100 and follow a bell curve, most people fall within the range of 85 to 115. The percentage of people who score above 100 is the same as those who score below it - about 50%.

According to the nature of a bell curve, approximately 34% of people score between 100 and 115, and about 14% score above 115. Therefore, around 50% (34% + 14%) of the people score above 100.

Applying this percentage to the population of the 78 students, we would expect about 50% of this student population to have scores above 100. So, 50% of 78 students amounts to 39 students. Rounded off to one decimal place, the answer would be 39.0 students. However, students can't be divided, so we conclude that about 39 students from the group of 78 have IQ scores above 100.

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Answer:

3.    

Step-by-step explanation:

the answer, by putting 1/3 in the place of x the answer comes out to 3.

The speed of Mercury around the Sun in kilometers/second is 48 kilometers/second.

In 2  seconds, Mercury will travel 96 kilometers.

The speed of an object is the ratio of the distance travelled by the object to the time taken by the object to travel that distance.

Given, the speed of Mercury around the Sun is 30 miles/second.

Therefore, the speed of Mercury around the Sun in kilometers/second

= (30 × 1.6) kilometers/second

= 48 kilometers/second

Now, in 2  seconds, Mercury will travel

= 48 × 2 kilometers

= 96 kilometers

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Answer:

55.88

Step-by-step explanation:

To solve (679 × 0.0025) + 24.17, multiply 679 by 0.0025 to obtain 1.6975, then add this to 24.17 to get the final result, which rounded to the nearest thousandth is 25.868.

To find the best answer to report for the calculation (679 × 0.0025) + 24.17, we first perform the multiplication of 679 by 0.0025. This step is crucial to solve the problem accurately. After carrying out the multiplication, we add the result to 24.17 to find the final answer.

Calculating 679 × 0.0025 gives us 1.6975. The next step is to add this product to 24.17:

1.6975 + 24.17 = 25.8675

Thus, the best answer to report, rounded to the nearest thousandth as is common in many mathematical contexts, would be 25.868.

Complete Question:

What is the best answer to report for: (679 × 0.0025) + 24.17?

Answer:

[tex]\Rightarrow 6\times 10^5\ g[/tex]

Step-by-step explanation:

Given : Divide two number and write as scientific notation.

Scientific notation:

[tex]\Rightarrow k\times 10^n[/tex]

Number 1: 300 g²

Number 2: 0.0005 g

Divide [tex]=\dfrac{300\ g^2}{0.0005\ g}[/tex]

[tex]\Rightarrow \dfrac{30\times 10}{5\times 10^{-4}\ g{2-1}[/tex]

[tex]\Rightarrow 6\times 10^{1+4}\ g[/tex]

[tex]\Rightarrow 6\times 10^5\ g[/tex]

Scientific notation:

[tex]\Rightarrow k\times 10^n[/tex]

Hence, The result of division is [tex]\Rightarrow 6\times 10^5\ g[/tex]