Which are the possible side lengths of a triangle?
3 cm, 5 cm, 9 cm
4 cm, 8 cm, 10 cm
6 cm, 9 cm, 17 cm
8 cm, 10 cm, 18 cm

Answer: D = -16

Step-by-step explanation: First you have to isolate the variable by subtracting the coefficient 8D from both equations then subtracting 1 from both equations to isolate 1D.

9d + 1 = 8d - 15

1d + 1 = -15

1d = -16

D = -16

hope this helped

Answer: The Answer To 9d + 1 = 8d -15

Step-by-step explanation:

STEP 1. Combine Like Terms As Well As Changing The Sign(s)

(WHEN YOU CHANGE SIDES YOU CHANGE THE SIGNS!!!)

STEP 2. Switch Signs Or Make It Opposite

d + 1 = - 15

Answer:

x = - [tex]\sqrt{7}[/tex]

Step-by-step explanation:

Radical roots occur in pairs, that is

x = [tex]\sqrt{7}[/tex] is a root then so is x = - [tex]\sqrt{7}[/tex]

Answer:

-√7 = -2.64

Step-by-step explanation:

The polynomial function has roots. The first root is 3 and the second is √7.

When we have a square root that means that we get two roots from the same number but one is negative and the other is positive. For example, if we have:

√x² = ±x

Because we can have:

(-x)² = x², or

(x)²=x².

So a square root always gives us two answers, one negative and the other positive.

Answer:

233.33 %.

Step-by-step explanation:

The increase is 10 - 3 = 7c.

Percentage increase =  100 * 7 / 3.

= 700 / 3

=  233.33 %.

The cost of a first-class postage stamp increased approximately 233.33% from 1917 to 1974.

To calculate the percent increase in the cost of a first-class postage stamp from 1917 to 1974, follow these steps:

Thus, the percent increase in the cost of a first-class postage stamp from 1917 to 1974 is approximately 233.33%.

Answer:

x(1 - .4)

Step-by-step explanation:

x = regular price.

1 - .4 = .6 = 60%

The sale price is equal to the full price (aka x) minus the discounted price (40% of x = 40/100 times x =  .4x)

Therefore sale price = x - .4x or x(1 - .4)

Answer:

There are 108900 different committees can be formed

Step-by-step explanation:

* Lets explain the combination

- We can solve this problem using the combination

- Combination is the number of ways in which some objects can be

 chosen from a set of objects

-To calculate combinations, we will use the formula nCr = n!/r! × (n - r)!

 where n represents the total number of items, and r represents the

 number of items being chosen at a time

- The value of n! is n × (n - 1) × (n - 2) × (n - 3) × ............ × 1

* Lets solve the problem

- There are 12 men and 12 women

- We need to form a committee consists of 3 men and 4 women

- Lets find nCr for the men and nCr for the women and multiply the

 both answers

∵ nCr = n!/r! × (n - r)!

∵ There are 12 men we want to chose 3 of them

∴ n = 12 and r = 3

∴ nCr = 12C3

∵ 12C3 = 12!/[3!(12 - 3)!] = 220

* There are 220 ways to chose 3 men from 12

∵ There are 12 women we want to chose 4 of them

∴ n = 12 and r = 4

∴ nCr = 12C4

∵ 12C4 = 12!/[4!(12 - 4)!] = 495

* There are 495 ways to chose 4 women from 12

∴ The number of ways to form different committee of 3 men and 4

   women = 220 × 495 = 108900

* There are 108900 different committees can be formed

Answer:

sin(45°)= AC/9

cos(45°)= BC/9

Step-by-step explanation:

This is a right angle triangle:

∠ABC =∠CAB = 45°

Now

AC= CB

AB = 9 units.

We will apply sines:

sine(45°)= AC/AB

We know that AB = 9 units.

So substitute the value of side AB

sin(45°)= AC/9

Now apply cos(45°)

cos(45°)=  BC/AB

Again substitute the value of AB:

cos(45°)= BC/9

Thus the answer is

sin(45°)= AC/9

cos(45°)= BC/9  ....

Answer:

A

E

Step-by-step explanation:

Answer:

B

Step-by-step explanation:

Given the 2 equations

y = 2x² - 5x - 7 → (1)

y = 2x + 2 → (2)

Since both equations express y in terms of x we can equate the right sides, that is

2x² - 5x - 7 = 2x + 2 ( subtract 2x + 2 from both sides )

2x² - 7x - 9 = 0 ← in standard form

Consider the factors of the product of the x² term and the constant term which sum to give the coefficient of the x- term

product = 2 × - 9 = - 18 and sum = - 7

The factors are + 2 and - 9

Use these factors to split the x- term

2x² + 2x - 9x - 9 = 0 ( factor the first/second and third/fourth terms )

2x(x + 1) - 9(x + 1) = 0 ← factor out (x + 1) from each term

(x + 1)(2x - 9) = 0

Equate each factor to zero and solve for x

x + 1 = 0 ⇒ x = - 1

2x - 9 = 0 ⇒ 2x = 9 ⇒ x = 4.5

Substitute these values into (2) for corresponding values of y

x = - 1 : y = (2 × - 1) + 2 = - 2 + 2 = 0 ⇒ (- 1, 0)

x = 4.5 : y = (2 × 4.5) + 2 = 9 + 2 = 11 ⇒ (4.5, 11)

Solutions are (4.5, 11) and (- 1, 0)

Answer:

(-1, 0) , (4.5,  11). (the second choice).

Step-by-step explanation:

y = 2x^2 - 5x - 7

y = 2x + 2

Since both right side expressions are equal to y we can equate them.

2x^2 - 5x - 7 = 2x + 2

2x^2 - 7x - 9 = 0

(2x - 9)(x + 1)

x = 4.5 , -1.

Substituting these values of x in the second equation:

When x = -1 , y =2(-1) + 2 = 0.

When x = 4.5, y =  2(4.5) + 2 = 11.

The answer is:

It will take 42.35 minutes to weed the garden together.

To solve the problem, we need to use the given information about the rate for both Laura and her husband. We know that she can weed the garden in 1 hour and 20 minutes (80 minutes) and her husband can weed it in 1 hour and 30 minutes (90 minutes), so we need to combine both's work and calculate how much time it will take to weed the garden together.

So, calculating we have:

Laura's rate:

[tex]\frac{1garden}{80minutes}[/tex]

Husband's rate:

[tex]\frac{1garden}{90minutes}[/tex]

Now, writing the equation we have:

[tex]Laura'sRate+Husband'sRate=CombinedRate[/tex]

[tex]\frac{1}{80}+\frac{1}{90}=\frac{1}{time}[/tex]

[tex]\frac{1*90+1*80}{7200}=\frac{1}{time}[/tex]

[tex]\frac{170}{7200}=\frac{1}{time}[/tex]

[tex]\frac{17}{720}=\frac{11}{time}[/tex]

[tex]\frac{17}{720}=\frac{1}{time}[/tex]

[tex]\frac{17}{720}*time=1[/tex]

[tex]time=1*\frac{720}{17}=42.35[/tex]

Hence, we have that it will take 42.35 minutes to weed the garden working together.

Have a nice day!

Answer:

[tex]\sqrt{37}[/tex]

Step-by-step explanation:

Distance formula

[tex]d = \sqrt {\left( {x_1 - x_2 } \right)^2 + \left( {y_1 - y_2 } \right)^2 }[/tex]

[tex]d = \sqrt {\left( {1 - 2 } \right)^2 + \left( {-2 - 4 } \right)^2 }[/tex]

Simplify

[tex]d = \sqrt {\left( {-1 } \right)^2 + \left( {-6 } \right)^2 }[/tex]

Simplify

[tex]d = \sqrt {\left 1 + \left 36}[/tex]

[tex]d = \sqrt{37}[/tex]

Answer

[tex]d = \sqrt{37}[/tex]

Final Answer:

The derived relationship between p and q that is independent of y is: q = 1/2 * p

Explanation:

To find the relation between 'p' and 'q' that is independent of 'y,' we will combine the two given equations and eliminate 'y'.
The equations given are:
1) -2y² - 2y = p
2) -y² + y = q
First, we want to manipulate these equations to isolate similar terms. Notice that the first equation has -2y² and the second has -y². If we multiply every term in the second equation by 2, we will have a coefficient of -2y² in the second equation, which will help us cancel out the y² terms. Let's do that:
2(-y² + y) = 2q
-2y² + 2y = 2q
Now, let's subtract the second equation from the first equation:
(-2y² - 2y) - (-2y² + 2y) = p - 2q
On subtracting, -2y² will cancel out with -2y², and -2y will subtract 2y to give -4y:
-2y² + 2y² - 2y - 2y = p - 2q
0 - 4y = p - 2q
-4y = p - 2q
Since we want a relationship without 'y', we can't do much with this result directly, as it still contains 'y'. But let's look at the equations we've been given once more.
The goal is not to solve for 'y' but to find a relationship between 'p' and 'q'. To accomplish this, let's compare the two original equations and try to eliminate 'y' by dividing them. Divide the first equation by the second equation:
(-2y² - 2y) / (-y² + y) = p / q
Now, factor out -y from both the numerator and the denominator:
- y(2y + 2) / - y(y - 1) = p / q
Simplify the expression by canceling out the -y term:
(2y + 2) / (y - 1) = p / q
At this point, you can see that there is no straightforward way to solve this for a relationship that is completely independent of 'y' because the y's don't cancel out.
One method to proceed, since we must get rid of 'y', is to compare coefficients that correspond to the same powers of 'y' assuming p and q are related through such a power series.
We have from the first equation by rearranging:
y² + y = -p/2
Comparing coefficients to the second equation:
y² = -q
y = q
By matching coefficients for the same powers of y, we deduce:
y (from -y²) = -q (from -y² + y), so q = 1/2 * p
Thus, our derived relationship between p and q that is independent of y is:
q = 1/2 * p
This indicates that q is half of p.

Answer:

40

Step-by-step explanation:

Subtracting a number k from 300 looks like this 300-k.

Adding a number k to 220 looks like this 220+k.

They are saying for some number k that we have 300-k and 220+k is the same value.

That is, 300-k=220+k.

This is the equation we are going to solve for your number.

300-k=220+k

Add k on both sides:

300=220+2k

Subtract 220 on both sides;

80=2k

Divide both sides by 2:

40=k

k=40.

So the number is 40.

Check: 300-40=260 while 220+40=260.

Answer:

2) y-x=2

Add x on both sides:

 y        =x+2

The slope is 1 and the y-intercept is 2.

Step-by-step explanation:

So a linear equation in slope-intercept form is y=mx+b where m is the slope and b is the y-intercept.

Let's put all of these in that form:

1) y+x=2

Subtract x on both sides:

 y     =-x+2

The slope is -1 and the y-intercept is 2.

2) y-x=2

Add x on both sides:

 y        =x+2

The slope is 1 and the y-intercept is 2.

3) y-x+2=0

Add x on both sides:

 y     +2=x

Subtract 2 on both sides:

 y        =x-2

The slope is 1 and the y-intercept is -2.

4) y+x-2=0

Add 2 on both sides:

 y +x      =2

Subtract x on both sides:

 y           =-x+2

The slope is -1 and and the y-intercept is 2.

Answer:

C. Marc

Step-by-step explanation:

to get the amount they earn in an hour divide the amount they earn by the amount of hours they worked for.

Maria: 60 divided by 8 = 7.50

Marc: 46.50 divided by 6 = 7.75

Answer:

D.f(x) = −4x + 105

Step-by-step explanation:

Since the function in linear, we know it has a slope.

We know 2 points

(5,85) and (10,65) are 2 points on the line

m = (y2-y1)/(x2-x1)

   = (65-85)/(10-5)

    =-20/5

    =-4

We know a point and the slope, we can use point slope form to write the equation

y-y1 =m(x-x1)

y-85 = -4(x-5)

Distribute

y-85 = -4x+20

Add 85 to each side

y-85+85 = -4x+20+85

y = -4x+105

Changing this to function form

f(x) =-4x+105

Answer: D or f(x) = -4x + 105

Answer:

D

Step-by-step explanation:

For a sequence to be an arithmetic sequence, it must have a common difference. In other words, it must either go down by the same number or up by the same number.

Let's look at the choices:

A. {-10,5,-2.5,1.25,...}

-10 to 5, that went up by 15. It has to keep going up by 15 to be arithmetic. However 5+15 is not -2.5 so it isn't arithmetic.

B. {100,20,4,0.8,...}

100 to 20, that went down by 80. Since 20-80 is not 4, then this sequence is not arithmetic.

C. {1,4,16,48,...}

1 to 4, that went up by 3. 4+3 is not 16 so this is not arithmetic.

D. {-10,-3.5,3,9.5,...}

-10 to -3.5, that went up 6.5.

-3.5+6.5=3

3+6.5=9.5

This is arithmetic. It keeps going up by 6.5.

Answer:

{2, 3, √13}  

Step-by-step explanation:

In a right triangle, the sum of the squares of the two shorter sides equals the square of the third side (Pythagoras).

Let's check each set of sides in turn.

A. {2, 3, √13}

2² + 3² = 4 + 9 = 13

(√13)² = 13

This is a right triangle.

B. {5, 5, 2, √10}

This is a quadrilateral (four sides).

C. {5, 12, 15}

5² + 12² = 25 +144 = 169

15² = 225

This is not a right triangle.

Option B, 1/3

Since pepper topping can be in any size pizza, it remains as a 1/3 probability, since there are three toppings per size.

The logarithm function [tex]\log_ab[/tex], where [tex]a,b>0 \wedge a\not =1[/tex], is increasing for [tex]a\in(1,\infty)[/tex] and decreasing for [tex]a\in(0,1)[/tex]

[tex]\ln x =\log_ex[/tex] and [tex]e\approx 2.7>1[/tex] therefore [tex]\ln x[/tex] is increasing.

Answer:

[tex]\frac{x-2}{x^2-x-2}[/tex]

Step-by-step explanation:

A removable discontinuity is when there is a hole in your graph. This is usually because one X value has been canceled out. Most of the time, it takes factoring to figure out if there is a removable discontinuity when looking at an equation.

First, look at the numerator [tex]x-2[/tex] . This can't be factored any further. However, [tex]x^2-x-2[/tex] can be factored since it is a trinomial (has three terms) .

For the purposes of this example, you may want to think about it as

[tex]1x^2 -1x-2[/tex]

To factor, multiply the the outside coefficients

1 x -2 = -2

Now take the middle coefficient (-1) and ask yourself what two numbers multiply to make -2, but still add to be -1.

-2 x 1 = -2

-2 + 1 = -1

So in factored form, the equation is

[tex]\frac{x-2}{(x-2)(x+1)}[/tex]

Since you have x-2 on both top and bottom, that can be canceled out. x - 2 would be your removable discontinuity in this situation.

A removable discontinuity can occur in a function if there are common factors in both the numerator and denominator that can be canceled out.

A function has a removable discontinuity at a particular point if the function is undefined at that point but can be made continuous by redefining the value at that point. To identify the removable discontinuity, we need to factor both the numerator and denominator of the function. By factoring, we can determine if any common factors exist that can be canceled out, resulting in a removable discontinuity.

Let's consider the given functions:

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Answer:

a. a correlation matrix

Step-by-step explanation:

The best fitting straight line for a data set of X-values plotted against y-values is  called a correlation matrix.

Step-by-step explanation:

Write a proportion:

2½ cups / 1 batch = x cups / 3¾ batches

Cross multiply:

x × 1 = 2½ × 3¾

To multiply the fractions, first write them in improper form:

x = (5/2) × (15/4)

x = 75/8

Now write in proper form:

x = 9⅜

Your answer is correct!  Well done!

Answer:

The rule is y = 4x - 5.

Step-by-step explanation:

Notice that if we start with x = 1 and increase x by 1, we get 2.  Simultaneously, y starts with -1 and becomes 3.  Thus, the slope is m = rise / run = 4/1, or 4.

The rule is y = 4x - 5.

Check:  Suppose we pick input 4 from the table. Does this rule produce output 11?  Is 11 = 4(4) - 5 true?  YES.

Answer:

y = 4x - 5

Step-by-step explanation:

Have you been taught to set up 2 equations and 2 unknowns?

That is actually the only way I could do this.

y = mx + b

x = 2

y = 3

3 = 2m + b

x = 1

y = -1

-1 = m + b        Multiply by 2. That means that the m term will cancel.

================

-2 =2m +2b    

3 = 2m + b        Subtract

-5 = b

==================

3 = 2m + b       Substitute - 5 for b

3 = 2m - 5        Add 5 to both sides.

3+5= 2m-5+5   Combine

8 = 2m               Divide by 2

8/2=2m/2

m = 4

Answer:

Approximately 92.51.

Not sure what the desired rounding is since it isn't listed.

Step-by-step explanation:

So the first vector G is 40.3 m long in a -35 degree direction.

Lat's find the components of G.

[tex]G_x=40.3\cos(-35)=33.0118[/tex].

[tex]G_y=40.3\sin(-35)=-23.1151[/tex].

The second vector H is 63.3 m long in a 270 degree direction.

[tex]H_x=63.3\cos(270)=0[/tex].

[tex]H_y=63.3\sin(270)=-63.3[/tex].

The resultant vector can be found by adding the corresponding components:

[tex]R_x=G_x+H_x=33.0118+0=33.0118[/tex]

[tex]R_y=G_y+H_y=-23.1151+(-63.3)=-86.4151[/tex]

Now we are asked to find the magnitude of [tex](R_x,R_y)[/tex] which is given by the formula [tex]\sqrt{R_x^2+R_y^2}[/tex].

Since [tex](R_x,R_y)=(33.0118,-86.4151)[/tex] then the magnitude is [tex]\sqrt{(33.0118)^2+(-86.4151)^2}=\sqrt{8557.34844}=92.51[/tex].

The magnitude of the sum of vector G (40.3m, -35°) and vector H (63.3m, 270°) is found by breaking each vector into its components, summing these components, and using the Pythagorean theorem. The magnitude of the sum of these vectors is approximately 92.1 m.

Given that vector G has a magnitude of 40.3 m and is in a -35.0° direction, and vector H has a magnitude of 63.3 m and is in a 270° direction, the sum of these vectors can be determined. This sum is found by breaking each vector into its component forms, adding the components together, and then using the Pythagorean theorem to find the magnitude of the result.

For vector G: Gx = 40.3m * cos(-35) = 33m and Gy = 40.3m * sin(-35) = -23.14m. For vector H: Hx = 0 (as sin(270) equals 0) and Hy = -63.3m (as sin(270) equals -1). The sum vector S = (Gx+Hx, Gy+Hy) = (33m+0 , -23.14m-63.3m) = (33m, -86.44m). Thus, to find the magnitude of the sum of the vectors, we use the Pythagorean theorem: |S| = sqrt((33m)² + (-86.44m)²) = 92.1 m (rounded to 1 decimal place).

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Answer:

You can use teh associative property to split 48 and 82 each into 2 peices

(40+8)(80+2) then you can move the parenthesis around. 40+(8+80)+2

40+(88+2)

40+90=130

[tex]\text{Hey there!}[/tex]

[tex]\text{a = m}^2[/tex]

[tex]\text{If m = -3 replace the m-value in the problem with -3}[/tex]

[tex]\text{a = -3}^2[/tex]

[tex]\huge\text{-3}^2\text{ = -3 * 3 = -9}[/tex]

[tex]\boxed{\boxed{\huge\text{Answer: a = -9}}}\huge\checkmark[/tex]

[tex]\text{Good luck on your assignment and enjoy your day!}[/tex]

~[tex]\frak{LoveYourselfFirst:)}[/tex]

Oil added or consumed during the day equals evening oil level minus morning oil level (g - h).

To calculate how much oil was consumed or added during the day, you can use the formula:

Change in oil = g - h

If the value is positive, it means oil was added.

If the value is negative, it means oil was consumed (leaked or used).

For example, if in the morning your car had 5 liters of oil (h = 5) and in the evening it had 7 liters (g = 7):

Change in oil = 7 - 5 = 2 liters

This means 2 liters of oil were added during the day.

Answer:

b

Step-by-step explanation:

Given

3x² - 18x = 21 ( divide all terms by 3 )

x² - 6x = 7

To complete the square

add ( half the coefficient of the x- term )² to both sides

x² + 2(- 3)x + 9 = 7 + 9

(x - 3)² = 16 ( take the square root of both sides )

x - 3 = ± [tex]\sqrt{16}[/tex] = ± 4 ( add 3 to both sides )

x = 3 ± 4, hence

x = 3 - 4 = - 1 and x = 3 + 4 = 7

The correct option is b. [tex]\((x-3)^2=16\); -1 and 7.[/tex]

To solve the given quadratic equation [tex]\(3x^2 - 18x = 21\),[/tex] we first divide the entire equation by 3 to simplify it:

[tex]\[ x^2 - 6x = 7 \]\[ x^2 - 6x + 9 - 9 = 7 \] \[ (x - 3)^2 - 9 = 7 \][/tex]

Now, we isolate the perfect square on one side:

[tex]\[ (x - 3)^2 = 7 + 9 \] \[ (x - 3)^2 = 16 \][/tex]

This is the perfect square equation. To find the solutions, we take the square root of both sides:

[tex]\[ x - 3 = \pm4 \][/tex]

Now, we solve for \(x\) by adding 3 to both sides:

[tex]\[ x = 3 \pm 4 \][/tex]

This gives us two solutions:

[tex]\[ x = 3 + 4 = 7 \] \[ x = 3 - 4 = -1 \][/tex]

 Therefore, the solutions to the equation are [tex]\(x = -1\) and \(x = 7\),[/tex] which corresponds to option b. [tex]\((x-3)^2=16\); -1 and 7.[/tex]

Answer:

[tex]y=2x-48[/tex]

Step-by-step explanation:

Let

y -----> the profit earned by the hot dog stand daily

x ----> the number of hot dogs sold

we know that

The linear equation that represent this problem is equal to

[tex]y=2x-48[/tex]

This is the equation of the line into slope intercept form

where

[tex]m=2\frac{\$}{hot\ dog}[/tex] ----> is the slope

[tex]b=-\$48[/tex] ---> is the y-intercept (cost of the day's supply)

The question relates to the linear function concept. In context of the problem, the profit earned by the hot dog stand is represented by the equation y = 2x - 48, where 'y' is the profit, 'x' is the number of hot dogs sold, '2' is the profit per hot dog, and '48' is the fixed daily cost.

The question relates to the concept of a linear function in Mathematics. In this case, the profit (y) made by the hot dog stand depends on the number of hot dogs sold (x). The stand has a fixed cost of $48 for each day's supply, and then makes a profit of $2 for each hot dog sold.

The linear function can be represented by the equation y = mx + b, where 'm' is the slope of the line (representing the rate of profit per hot dog sold, which is $2), 'x' is the number of hot dogs sold, and 'b' is the y-intercept (representing the fixed costs of the stand, which is -$48).

Therefore, the equation representing the profit of the hot dog stand for x number of hot dogs sold is: y = 2x - 48.

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Answer:

C) f(x) = x2

Step-by-step explanation:

[tex]\bf \textit{Logarithm Cancellation Rules} \\\\ log_a a^x = x\qquad \qquad a^{log_a x}=x \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \log_6(\sqrt[3]{6})\implies \log_6(6^{\frac{1}{3}})\implies \cfrac{1}{3} \\\\[-0.35em] ~\dotfill\\\\ \log_2(64)\implies \log_2(2^6)\implies 6 \\\\[-0.35em] ~\dotfill\\\\ -3\log_5(25)\implies -3\log_5(5^2)\implies -3(2)\implies -6 \\\\[-0.35em] ~\dotfill[/tex]

[tex]\bf \log_2(\sqrt[4]{8})\implies \log_2(\sqrt[4]{2^3})\implies \log_2(2^{\frac{3}{4}})\implies \cfrac{3}{4} \\\\[-0.35em] ~\dotfill\\\\ \log_3\left( \frac{1}{81} \right)\implies \log_3\left( \frac{1}{3^4} \right)\implies \log_3(3^{-4})\implies -4[/tex]

Answer:

3/4 goes with [tex]\log_2(8^\frac{1}{4})[/tex]

-4 goes with [tex]\log_3(\frac{1}{81})[/tex]

-6 goes with [tex]-3\log_5(25)[/tex]

1/3 goes with [tex]\log_6(6^\frac{1}{3})[/tex]

Step-by-step explanation:

[tex]\log_6(6^\frac{1}{3})=\frac{1}{3}\log_6(6)=\frac{1}{3}\cdot 1=\frac{1}{3}[/tex]

[tex]\log_2(64)=6 \text{ since } 2^6=64[/tex]

[tex]-3\log_5(25)=-3(2)=-6 \text{ since } 5^2=25[/tex]

[tex]\log_2(8^\frac{1}{4})=\frac{1}{4}\log_2(8)=\frac{1}{4}\log_2(2^3)=\frac{1}{4}\cdot (3)\log_2(2)=\frac{1}{4} \cdot 3 \cdot 1=\frac{3}{4} [/tex]

[tex] \log_3(\frac{1}{81})=\log_3(\frac{1}{3^4})=\log_3(3^{-4})=-4\log_3(3)=-4(1)=-4[/tex]

Here are few rules I used:

[tex]\log_a(b)=x \text{ means } a^x=b[/tex]

[tex]\log_a(a)=1 [/tex]

[tex]\log_a(b^r)=r \log_a(b)[/tex]