Mathematics High School

## Answers

**Answer 1**

**Answer:**

Below

**Step-by-step explanation:**

Simple interest = P r t P = 150 principal r = decimal interest rate = .06

t = time

so re-arrange...

interest/(Pr) = t Plug in the given numbers

63 / ( 150*.06) = t = **7 years**

## Related Questions

Graph y = 2 cos (x + pi/4)

### Answers

The required **graph **has been attached below that represents the **function **y = 2 cos (x + π/4).

What is the cosine function?

The cosine function has a range of values between -1 and 1, and it is a **periodic function **that repeats every 2π radians or 360 degrees. The graph of the cosine function is a wave-like **curve **that oscillates between -1 and 1, with a **period** of 2π radians or 360 degrees.

To plot the graph of the **function **y = 2 cos (x + π/4), we can follow these steps:

The coefficient of the **cosine **term is 2, so the amplitude is 2.

The coefficient of the **angle **x is 1, so the period is 2π.

The **constant term **is π/4, so the phase shift is -π/4.

The y-**intercept **occurs when x = 0, which gives y = 2 cos (π/4) = √2.

Using the amplitude, period, phase shift, and intercepts, sketch the graph of the **function **over one period. Then, repeat the graph every 2π to get the complete **graph**.

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8. Express each of the side lengths

of triangle PQR as a square root.

9. Check for Reasonableness Estimate

the perimeter of triangle PQR. Round

to the nearest hundredth.

-8-6

P(-5,-2)

R(-1,3)

4

y

-2 O

2

4

X

Q(2,-2)

### Answers

Therefore , the solution of the given problem of** triangle **comes out to be this indicates that triangle PQR's circumference is roughly 22.73 units.

What is a triangle?

A triangle is a **polygon **because it has four or thereabouts additional parts. It has a simple rectangle shape. A rectangle with sides A, B, and C is referred to as a triangle. When the sides are actually not collinear, Euclidean **geometry **yields a singular plane and cube. If a triangular has three parts and three angles, it is a polygon.

Here,

We can determine the length of each edge of triangle PQR using the distance formula:

Front PQ:

=>√[(2 - (-5))²+ (-2 - (-2)²]

=> √[(7)² + (0)²]

=> √(49)

=> 7

Slant QR:

=> √[(2 - (-1))² + (-2 - 3)²]

=> √[(3)² + (-5)²]

= > √(9 + 25)

= √(34) (34)

Slant RP:

=> √[(-1 - (-5))² + (3 - (-2))²]

=> √[(4)² + (5)²]

= > √(16 + 25)

= > √(41) (41)

Therefore, the triangular PQR's side lengths are 7, 34, and 41.

We can sum the side lengths to determine the perimeter and then verify that it is reasonable:

=> 7 + √34 + √41 ≈ 22.73

This indicates that triangle PQR's circumference is roughly 22.73 units. Given the vertices' coordinates as well as the lengths of a sides written as square roots, this seems plausible.

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Answer:

Step-by-step explanation:

PLEASE HELP. WILL GIVE BRAINLIEST.

### Answers

The volume of water following in a **cylindrical** pipe every second is 1205.76 cubic inches.

What is the volume?

Volume is the measure of the **capacity** that an object holds.

Formula to find the volume of the object is Volume = Area of a base × Height.

Given that,

a) Dimensions of cylindrical pipe are height= 2 feet=24 inches and diameter = 8 inches.

Here, radius = 8/2 = 4 inches

We know that, **volume** of cylinder = πr²h

= 3.14×(4)²×24

= 1205.76 cubic inches

b) In 5 minutes

Volume of water filled in tank

= 1205.76×5×60

= 361728 cubic inches

A cylindrical tank with a diameter of 15 feet and a height of 6 feet.

Radius =7.5×12=90 inches

Now, 361728=3.14×(90)²×h

361728=25434h

h=361728/25434

h=14.2 inches

c) Volume of tank = 3.14×(90)²×72

= 1831248 cubic inches

75% of volume = 75% of 1831248

= 75/100 ×1831248

= 0.75×1831248

= 1373436 cubic inches

Time taken to fill 75% of tank = 1373436/1205.76×60

= 1373436/72345.6

= 19 minutes

Therefore, the volume of water following in a **cylindrical** pipe every second is 1205.76 cubic inches.

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ƒ(x)= -2/3x - 4

ƒ (9) =

If f(x)=0, then x = 0

Pls help!

### Answers

**Answer:**

-6,0 0,-4

**Step-by-step explanation:**

f(x)= -2/3x-4 = 0,-4

-2/3x-4 = -6,0

James Harden made 12 shots in a recent game for a total of 29 points. If Harden did not make any free throws, can we identify how many of his shots were 2-pointers? How many shots were 3-pointers?

This is a problem that has to do with substitution and elimination.

Thanks.

### Answers

**Answer:**

7 2-pointers and 5 3-pointers

**Step-by-step explanation:**

If it has to do with substitution/elimination. Sounds like a system of equations.

What’s number 12??????

### Answers

**Answer: C**

**Step-by-step explanation:**

**Option:-**

[tex] \large\tt{ \blue{c)3\pi}}[/tex]

**Ans:-**

**π** belongs to the **set** of **irrational numbers**. Therefore it is never true that **3π** is a rational number.The product of any **rational number** and **any irrational number** will always be an **irrational number.** This allows us to quickly conclude that **3π**** **is** ****irrational.**

**[tex] \: [/tex]**

hope it helps! :)

A teacher asks her students to write down the number of hours studied, rounded to the nearest half hour. She compiles the results and develops the probability distribution below for a randomly selected student. What is the mean of the probability distribution?

Probability Distribution

Hours Studied: X

Probability: P(X)

0.5

0.07

1

0.2

1.5

0.46

2

0.2

2.5

0.07

0.20

0.46

0.85

1.50

### Answers

**Answer:**

1.50

**Step-by-step explanation:**

To find the mean of the probability distribution, we need to multiply each value of hours studied by its corresponding probability, then add up the products.

Mean = (0.5 x 0.07) + (1 x 0.2) + (1.5 x 0.46) + (2 x 0.2) + (2.5 x 0.07)

Mean = 0.035 + 0.2 + 0.69 + 0.4 + 0.175

Mean = 1.5

Therefore, the mean of the probability distribution is 1.5.

**Answer:**

D

**Step-by-step explanation:**

1.50 on edge (if that guys right)

Jada likes to wear her hair in a high puff. She spends $12 on a package of 3 hair ties for natural hair. Which equation relates the cost , c , of the package of hair ties to the number of hair ties , h, in the package?

### Answers

4$ per hair tie so whatever letter it would be. The answer is 4$ per hair tie

Can anyone help?? PLEASE!!!!

### Answers

**Answer:**

yes, 2.79

**Step-by-step explanation:**

5$- 1.15(broccoli) = 3.85

3.85-1.06(2lemons)=2.79

**Answer: Yes Cameron will have money left over. He will have $2.79 left over.**

**Step-by-step explanation: 2 lemons = 1.06**

**1.06 + 1.15 = 2.21**

**5.00 - 2.21 = 2.79**

By the linear pair theorem,

is supplementary to

, which means

. It is given that

, so by the definition of congruent angles,

. Using the substitution property of equality, substitute

in for

to rewrite the previous equation as

. Thus

is supplementary to

by the definition of supplementary angles. By the linear pair theorem,

is supplementary to

. Since

and

are supplementary to

, then by the congruent supplements theorem,

.

Use the paragraph proof to complete the two-column proof.

What statement and reason belong in line 5?

### Answers

The **missing **statement is:

<1 is **supplementary **to <2 ( linear pair theorem)

What is congruent supplements theorem?

Congruent vertical angles exist. Congruent Additions Theorem: **Two angles **are **congruent** if they are **supplements** of the same angle, often known as congruent **angles**. Similar Complements Theorem: Two angles are congruent if they are the complements of the same angle, often known as congruent angles.

Given:

line and line b intersect (given)

<1 is supplementary to <2 ( **linear** pair theorem)

<3 is supplementary to <2 ( **linear pair **theorem)

Then, using **congruent** **supplements **theorem

<1 is supplementary to <2 and <3 is supplementary to <2 then

<3 is **supplementary **to <1

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What name is given to words that are formed to imitate sounds

### Answers

**Answer:**

Answer Below ↓

**Step-by-step explanation:**

Onomatopoeia is the formation of a word through imitation of a sound associated with that word.

are the vectors , , and linearly independent? linearly independent if they are linearly dependent, find scalars that are not all zero such that the equation below is true. if they are linearly independent, find the only scalars that will make the equation below true. .

### Answers

The vectors * u*,

*, and*

**v***are*

**w****linearly independent**if the only scalars that will make the equation

*true are*

**au + bv + cw = 0***,*

**a = 0***, and*

**b = 0***. If there are other scalars that are not all zero that will make the equation true, then the vectors are*

**c = 0****linearly dependent**.

To determine if the vectors are linearly independent or dependent, we can set up a matrix with the vectors as columns and use row operations to reduce it to reduced row echelon form. If the matrix has a pivot in every column, then the vectors are linearly independent. If there is a column without a pivot, then the vectors are linearly dependent.

Without the actual values of the vectors * u*,

*, and*

**v***, we cannot determine if they are linearly independent or dependent. If they are linearly dependent, we can find the scalars that are not all zero that will make the equation true by solving the system of equations represented by the matrix. If they are linearly independent, the only scalars that will make the equation true are*

**w***,*

**a = 0***, and*

**b = 0***.*

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Please help with the question in the picture

### Answers

The required **angle **in **radians **is approximately 3.33 radians.

What is arc?

**Arc **is defined as a **circular curve **drawn, where every point on the curve is equidistant from a fixed point.

Here,

The formula to find the angle in radians given the radius and arc length is,

angle (in radians) = arc length / radius

Substitute the values given,

angle = 30m / 9m = 3.33 radians (rounded to two decimal places)

Therefore, the **angle **in **radians **is approximately 3.33 radians.

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The temperature is 23 degrees F, at sunrise. By noon, the temperature changes by 10 degrees F. What could be the temperature at noon? The number line shows temperatures in degrees F. What are all the possible noon temperatures?

### Answers

The possible noon **temperatures** are 33 degrees F and 13 degrees F.

What is Equation?

Two or more expressions with an **Equal sign **is called as Equation.

If the **temperature** at sunrise is 23 degrees F and it changes by 10 degrees F by noon, then the temperature at noon could be:

23 + 10 = 33 degrees F

If we consider the temperature change as a positive or **negative** **deviation** from the sunrise temperature, then the temperature at noon could also be:

23 + (-10) = 13 degrees F

This means that the temperature has **decreased** by 10 degrees F from the sunrise temperature, resulting in a temperature of 13 degrees F at noon.

Therefore, another possible noon **temperature** is 13 degrees F.

Hence, the possible noon **temperatures** are 33 degrees F and 13 degrees F.

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Is 1/10 or 1/4 the larger fraction

### Answers

**Answer: **1/4 because if you have 1 pie cut into 4 pieces, each piece is going to be way bigger than if you cut a pie into 10 pieces, and took one of those pieces.

**Answer:**

1/4 is greater than 1/10 and the answer to the question "Is 1/4 greater than 1/10?" is yes

**Step-by-step explanation:**

Is 1/4 greater than 1/10? Is 1/4 bigger than 1/10? Is 1/4 larger than 1/10? These are all the same questions with one answer.

To get the answer, we first convert each fraction into decimal numbers. We do this by dividing the numerator by the denominator for each fraction as illustrated below:

1/4 = 0.25

1/10 = 0.1

Then, we compare the two decimal numbers to get the answer.

0.25 is greater than 0.1.

Therefore,1/4 is greater than 1/10 and the answer to the question "Is 1/4 greater than 1/10?" is yes

Which means that this equation is also true: 1/4 > 1/10

Note: When comparing fractions such as 1/4 and 1/10, you could also convert the fractions (if necessary) so they have the same denominator and then compare which numerator is larger.

Which answer choice gives three lengths that WILL form a triangle when put together?

### Answers

Using the side length property of the **triangle**, which is called the **triangle **inequality theorem, we found that **none **of the choices will form a **triangle**.

**What is 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. Any three locations that are not collinear provide a distinct **triangle **and a distinct plane.

The options given in the question are:

a)5,7,13

b) 12, 18, 35

c) 107,8,9

d) 4,5,10

One of the properties of the **length **of the sides of a **triangle **is as follows:

The sum of the **length **of any two sides of a **triangle **should be always greater than the third side.

Using this property, we can find which sides can be used to form a **triangle**.

a) 5 + 7 = 12

12 is not greater than 13.

So this cannot form a **triangle**.

b) 12+18 = 30

30 not greater than 35.

So this cannot form a **triangle**.

c) 8+9 = 17

17 not greater than 107.

So this cannot form a **triangle**.

d) 4+5 = 9

9 is not greater than 10.

So this cannot form a **triangle**.

Therefore using the side length property of the **triangle**, which is called the **triangle **inequality theorem, we found that **none **of the choices will form a **triangle**.

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an open box with a rectangular base is to be constructed from a rectangular piece of cardboard that is 16 inches by 21 inches. it is constructed by cutting a square from each corner and then bending up the resulting sides. find the size of the corner square (the square cut-out) that will produce a box having the largest possible volume. (disregard the thickness of the cardboard.)

### Answers

Using the quadratic formula, we find that x = 4 is the only positive, real solution. This means that the size of the corner square that will produce a box having the largest possible volume is 4 inches.

The size of the corner **square** that will produce a box having the largest possible volume is **4 inches**.

To find this, we need to first create an **equation **for the volume of the box in terms of the size of the corner square. Let's call the size of the **corner **square x.

The length of the box will be 16 - 2x (since we are **cutting **out x from each side of the length), the width will be 21 - 2x (since we are cutting out x from each side of the width), and the height will be x (since we are bending up the sides by the size of the corner square).

So the volume of the box is:

V = (16 - 2x)(21 - 2x)(x)

Next, we need to find the derivative of this equation in terms of x and set it equal to 0 to find the critical points:

V' = 4x^3 - 74x^2 + 336x

Setting this equal to 0:

4x^3 - 74x^2 + 336x = 0

Using the quadratic formula, we find that x = 4 is the only positive, real solution. This means that the size of the corner square that will produce a box having the largest possible volume is 4 inches.

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Let A = | | 1-1-1-1 | . Find a non-zero vector x in the null space of A.

### Answers

The null space of A is the set of all vectors of the form **[x₄, 2x₄, 3x₄, x₄]**, where x₄ is **not equal** to zero.

To solve this **equation**, we can use Gaussian elimination, which is a method for solving systems of linear equations. We start by augmenting the matrix A with the zero **vector **on the right-hand side, as follows:

| 1 -1 -1 -1 | 0 |

Next, we apply elementary row operations to the matrix to bring it into **row echelon** form. We can use the following row operations:

| 1 -1 -1 -1 | 0 |

| 0 -1 -2 -3 | 0 |

Then, we can express x₁, x₂, and x₃ in terms of x₄ as follows:

x₁ = x₄

x₂ = 2x₄

x₃ = 3x₄

Therefore, any vector of the form [x₄, 2x₄, 3x₄, x₄], where x₄ is not equal to zero, is a **non-zero** vector in the null space of A.

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The terminal sides of two angles of rotation intersect a circle centered at the origin at (-2, 3) and (3, "-2)" . How do the values of the sine and cosine of the angles compare? Explain.

### Answers

Comparing the **values **of sine and cosine for the two **angles**, we can see that:-

sin(θ₁) is positive while sin(θ₂) is negative.

cos(θ₁) is negative while cos(θ₂) is positive.

How to solve for rotation?

To **solve **this problem, we first need to **plot **the points (-2, 3) and (3, -2) on the **coordinate **plane to form the two **terminal **sides of the angles of rotation.

To find the values of **sine **and **cosine **for each **angle**, we can use the coordinates of the points where the terminal sides intersect the circle centred at the origin.

For angle θ₁:

x-coordinate of a point on circle = -2

y-coordinate of the point on circle = 3

Using the **Pythagorean theorem**, we can find the length of the hypotenuse of the right triangle formed by the terminal side of angle θ1 and the x-axis:

h² = (-2)²+ 3²

h² = 13

h = √13)

Therefore, we can find the values of sine and cosine for angle θ1 as follows:

sin(θ1) = y-coordinate / hypotenuse = 3 / √(13)

cos(θ1) = x-coordinate / hypotenuse = -2 / √(13)

For angle θ₂:

x-coordinate of a point on a circle = 3

y-coordinate of the point on the circle = -2

Using the Pythagorean theorem, we can find the length of the hypotenuse of the right triangle formed by the terminal side of angle θ₂ and the x-axis:

h₂ = 3₂ + (-2)₂

h²= 13

h = √(13)

Therefore, we can find the values of sine and cosine for angle θ2 as follows:

sin(θ₂) = y-coordinate / hypotenuse = -2 / √(13)

cos(θ₂) = x-coordinate / hypotenuse = 3 / √(13)

sin(θ₁) is positive while sin(θ₂) is negative.

cos(θ₁) is negative while cos(θ₂) is positive.

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Can someone help me with this please?

### Answers

**Answer:**

Tall = 32.9 in

**Step-by-step explanation:**

Pythagorean theorem:

diagonal² = width² + tall²

59² = 49² + tall²

3481 = 2401 + tall²

3481 - 2401 = tall²

1080 = tall²

√1080 = √tall²

tall = 32.863 in

The nearest thent is:

32.9 in

hello i am stuck at this question

### Answers

If you turn off the **spaceship's engine** and it continues to glide in a **straight line** in the direction it was when the engine was stopped, we first have to find the **speed** of the **spacecraft**, which can be given by the formula:

V(t)= (dx/dt, dy/dt) V(t)= (1, 2t-2)How could we calculate t=1 and t=2?

At **t=1**, the coordinates of the spaceship are (1,0). If we switch off the engine at this moment, the spaceship will continue to move in a **straight** **line** in the direction of its velocity vector, which is **(1,0)** at t=1.

At** t=2**, the coordinates of the spaceship are (2,1). If we switch off the engine at this moment, the spaceship will continue to move in a **straight** **line** in the direction of its velocity vector, which is **(1,2**) at t=2.

Therefore, at t=1 the spacecraft will move horizontally in a **straight line**. At t=2, the ship will move on a straight line with slope 2, passing through the point **(2,1).**

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In the formula p2 = mb/t, how does p change when m remains constant, b is doubled, and t is halved?

### Answers

In the formula** **p2 = mb/t, if m remains** constant**, b is **doubled**, and t is **halved**, then the value of p will increase.

This is because doubling b will **increase **the numerator of the fraction, while halving t will decrease the denominator of the fraction. This will result in** **a **larger value **for p2, and therefore a larger value for p.

To see this more clearly, let's plug in some numbers. Let's say that m = 2, b = 4, and t = 8. Plugging these values into the formula gives us:

p2 = (2)(4)/8 = 1

Now, let's double b and halve t. This gives us b = 8 and t = 4. Plugging these values into the formula gives us:

p2 = (2)(8)/4 = 4

As you can see, the value of p2 has increased from 1 to 4, which means that the value of p has also increased. Therefore, when m remains constant, b is doubled, and t is halved, the value of p will increase.

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if x/8=y/12, then x/y=

### Answers

If if x/8 = y/12, then x/y = 2/3., As **illustrated **below.

What is a numerical expression?

A numerical expression is a mathematical statement written in the form of numbers and unknown **variables**. We can form numerical expressions from statements.

Given, An expression [tex]\frac{x}{8} = \frac{y}{12}[/tex].

Now, If we move y to the **LHS **it will be on the denominator, and moving 8 to **RHS **will be placed on the numerator.

Therefore, [tex]\frac{x}{8} = \frac{y}{12}[/tex] is equal to [tex]\frac{x}{y} = \frac{8}{12}[/tex].

[tex]\frac{x}{y} = \frac{2}{3}[/tex].

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Why we should give importance to the filipinos who give honor for our country?

(ESP)

### Answers

It is important to give **importance** to the Filipinos who give honor to our country because they have sacrificed and dedicated their time and effort for the betterment of our nation.

These F**ilipinos** are the ones who have made a **significant impact** on our** country's progress** and development, and without them, our country would not be where it is today. Giving importance to these Filipinos also serves as a way to show our gratitude and appreciation for their contributions. Additionally, recognizing and honoring these Filipinos can inspire and motivate others to do the same and contribute to the betterment of our country. Therefore, it is essential to give importance to the Filipinos who give honor to our country.

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Need help on question 2 only!!

### Answers

**Step-by-step explanation:**

because of the specific graphic presentation (the 2 triangles together create a kite) we have more than the written information about the triangles.

the written information tells us only about one pair of corresponding sides that are equivalent (TS and TU).

but due to the graphic we know, that the central side VT is shared 1:1 by both triangles. so, we know about a second pair of corresponding sides that are equal (equivalent) : VT.

and we know that these are right-angled triangles, so via Pythagoras we know that also the 3rd sides (SV and UV) must be equal (equivalent).

so, we know via SSS (side-side-side) that the two triangles are equivalent.

3x +y = 3

2y = 10-6x

On a graph

### Answers

**Answer:**

GRAPH OF 3x + y = 3 (written on attachment)

GRAPH OF 2y = 10 - 6x (written on attachment)

1.2 John owns a special designed refuse truck. He hires out this truck at a daily rate of R5 000, excluding fuel. One person or more, depending on their numbers will decide on how to use the truck for the day to carry refuse to a nearby dumping ground. The graph on ANNEXURE A indicates the amount each person will pay depending on the number of people. Use the graph on ANNEXURE A and the above information to answer the questions that follow. 1.2.1 State the type of proportion represented in the graph on ANNEXURE A. 1.2.2 Write down the number of people in the group if each paid R625, 00. 1.2.3 Calculate the amount each person will pay if 9 people hired the truck.

### Answers

The **graph** does not represent a **proportional **relationship

The type of proportion represented in the graph

The complete question is added as an attachment

From the graph, we can see that the **points **on the line do not form a **straight line**

This means that the points are not **proportional**

The number of people in the group

The amount is given as

Amount = R625

From the graph, we have

People = **Approximately **5 when Amount = R625

So, the number of **people **is 5

The amount for 9 people

The number of people is given as

People = 9

From the **graph**, we have

People = 9 when Amount = R350

So, the amount is R350

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First to give me a full answer gets 100+

B, D, and F are the midpoints of each side and G is the centroid. Find the following lengths.

### Answers

**Step-by-step explanation:**

the point, where all medians intersect (G), is called centroid.

we use the centroid theorem :

the centroid of the triangle is at 2/3 of the distance (median) from the vertex to the opposite mid-point of the sides.

e.g. GE = 2/3 × BE

and that means the second (shorter) part of the median is the remaining 1/3 of the whole median.

1.

BG = 5 = 1/3 × BE

BE = 5×3 = 15

GE = 2/3 × BE = 2/3 × 15 = 10

2.

CG = 16 = 2/3 × CF

CF = 16×3/2 = 8×3 = 24

GF = 1/3 × CF = 24/3 = 8

3.

AD = 30

AG = 2/3 × AD = 2/3 × 30 = 20

GD = 1/3 × AD = 1/3 × 30 = 10

4.

GF = x = 1/3 × CF

CF = 3x

GC = 2/3 × CF = 2/3 × 3x = 2x

5.

AG = 9x = 2/3 × AD

GD = 5x - 1 = 1/3 × AD

therefore,

AG = 2 × GD

9x = 2(5x - 1) = 10x - 2

2 + 9x = 10x

2 = x

GD = 5x - 1 = 5×2 - 1 = 9 = 1/3 × AD

AD = 9×3 = 27

Alex surveyed twenty boys from his class. He asked their height in centimeters and their

shoe size. Alex wanted to know if these two variables were related. He graphed the

results on the coordinate plane. Each graph below shows Alex’s data.

Which of the graphs shows the BEST linear fit for the data?

### Answers

The **graph** that shows the best linear fit for the data is graph B.

What is graph?

Graph is a data structure composed of nodes **connected by edges**, which represent a relationship between the two notes of are used to represent networks of communication that organized and even extract concept like thought or riders the nodes in the graph are often represent by Circle and it's by line but the other safe of the **symbol** are also use copper power Bluetooth for a organizing and understanding data and can be used to solve complex problem.

Graph B shows a line that passes through the majority of the data points, indicating a strong positive correlation between the two variables. This means that as the height in centimeters increases, the shoe size also tends to increase. Graph A shows a less distinct line that doesn't pass through as many points as graph B, indicating a weaker correlation. Graph C does not show a linear fit, as the data points are scattered throughout the graph.

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Pick the ones that apply please

### Answers

The** true statements** about the **logarithm **are the ones in the third and fourth options.

Which statement is true about the logarithm?

Remember that a **logarithm **with a base can be written as a **quotient **between the natural logarithm of the argument and the natural logarithm of the base:

logₐ(b) = ln(b)/ln(a)

Then the given expression:

log₇(59) can be rewritten into:

ln(59)/ln(7)

So the last option is true.

And also, notice that log₇(59) = x means that 7^x = 59

Then if:

7^2 = 49

7^3 = 343

x must be between 2 and 3, so the third statement is also true.

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