Prove that log(3) 7 is an irrational number

Answers

Answer 1

Answer:

Proof below

Step-by-step explanation:

Irrational and Rational Numbers

We need to use the basic property of logarithms:

[tex]log_b\ a=x\ \ \Rightarrow b^x=a[/tex]

We are given:

[tex]log_3\ 7[/tex]

And we will prove it's an irrational number, i.e., it's not a rational number.

Recall a rational number can always be expressed as a fraction a/b, where a and b are integer numbers. Thus, if the expression was a rational number, then:

[tex]\displaystyle log_3\ 7=\frac{a}{b}[/tex]

Applying the property:

[tex]\displaystyle 3^{\frac{a}{b}}=7[/tex]

Raising to the power of b:

[tex]3^a=7^b[/tex]

It's not possible to find two integers a and b that satisfy the condition above. That contradiction comes from the assumption that [tex]log_3\ 7[/tex] is rational, thus, it's proven [tex]log_3\ 7[/tex] is irrational.


Related Questions

Which is equivalent to (x+3)(x-5)

Answers

Answer:

x^2 -2x -15

Step-by-step explanation:

by multiplying the 2 brackets

x^2

-5x + 3x = -2x

3 * -5= -15

16y5+4y5-12y3+15y3?
What is the simplified form?

Answers

Answer:

[tex]20y^5+3y^3[/tex]

Step-by-step explanation:

[tex]16y^5 + 4y^5 -12y^3 + 15y^3[/tex]

All we can do here is combine like terms (add those that have the same the same exponent)

[tex]20y^5+3y^3[/tex]

Good luck!

Answer:

[tex]20y^{5} +3y^{3}[/tex]

Step-by-step explanation:

[tex]16y^{5} +4y^{5} =20y^{5}[/tex] and [tex]-12y^{3} +15y^{3} =3y^{3}[/tex]

So [tex]20y^{5} +3y^{3}[/tex]

Help me with this I need help

Answers

23.3
Can I get a brainlist

Jared bakes 2 apple pies. He cuts the two pies into pieces. Each piece is 1/8 of a pie. Enter the number of pie Jared cuts.

Answers

Answer:

16 pieces

Step-by-step explanation:

Please give me brainliest

this is so emaruballos can be marked brainliest and yes the answer is 16

24 coloring books,60 crayons, and 84 markers can be packaged into at most how many Identical packages? How many will each package contain?

Answers

Answer:

there can be 12 identical packages at most. each package will contain:

- 2 coloring books

- 5 crayons

- 7 markers

Step-by-step explanation:

Please answer this correctly fast!! What type of triangle is this?
Options:
- SSS
- SAS
- ASA
- AAS
- HL
- Not Possible

Answers

Answer:

Not possible

Step-by-step explanation:

These triangles are not congruent

A school is selling t-shirts and hoodies to raise money. Cara bought a t-shirt and 2 hoodies for $60. Jack bought 3 t-shirts and one hoodie for $60. How much is one t-shirt and one hoodie?

Answers

One t-shirt is 12$ and one hoodie is 24$

SHOW
YOUR
WORK!!
X -
-67 V6 ²-4ac
2a
The
equation d = n²-12n + 43
models the number of defective
items a produced in a manufacturing process where there are
workers in a restricted
area
a) solve for
n
When
d=30
INTL
DLL

Answers

Answer:

1.20, 10.8

Step-by-step explanation:

d = n^2 - 12n + 43

d = 30

30 = n^2 - 12n + 43

0 = n^2 - 12n + 13

now solve for n using quadratic formula:

a=1, b=-12, c=13

n = (-b +- sqrt( b^2 -4ac))/2a

two answers ( one for + the square root, one for - the square root)

n = 1.20, n = 10.8

What’s 3/8 OF 5 as a mixed number

Answers

Answer:

Step-by-step explanation:

2 3;8

Answer:

1 7/8.

Step-by-step explanation:

5 * 3/8

= 15/8

= 1 7/8.

please help me step by step ​

Answers

Answer:

should be just be like doubling everytime

Step-by-step explanation:

6 water and 4 powder

12 water 8 powder

18 water 12 powder

it's the same ratio so it should taste the same, like 3 water and 2 powder is going to taste the same as 18 water and 12 powder

what is the answer for this( 7 3 x ∛27 - 2) x 1 5 + √81

Answers

Answer:

5208

Step-by-step explanation:

(73 x ³√27 - 2) x 15 + √81

(73 x 3 - 2) x 15 + 9

(219 - 2) x 24

217 x 24

5208

solve the following equation-7x-4y=-4 -9x-5y=-6

Answers

Answer:

[tex]\boxed{x = 4~~and~~ y = -6}[/tex]

.

Step-by-step explanation:

-7x - 4y = -4 (times 5) -35x - 20y = -20

-9x - 5y = -6 (times 4) -36x - 20y = -24

.

Eliminate both equations

[tex]-35x - 20y = -20[/tex]

-36x - 20y = -24    

[tex]x = 4[/tex]

.

Substitute the value of x to one of equations

[tex]-7x - 4y = -4[/tex]

[tex]-7(4) - 4y = -4[/tex]

[tex]-28 - 4y = -4[/tex]

[tex]-4y = - 4 + 28[/tex]

[tex]-4y = 24[/tex]

[tex]y = \frac{24}{-4}[/tex]

[tex]y = -6[/tex]

Step-by-step explanation:

ib2j2nwj wjwjw wjw w wiw w akwbysks soshsiw sksns sks sos sisIyah lah s ais ssj sj

A repair man charges $75.00 to come to the house as a service fee. The repairman charges $150 dollars per hour. Which equation best represents the total cost, c, of the repairs?


A. c = 150f + 2.75

B. c = 150-75f

C. c = 150f + 75

D. c = 225f

Answers

the answer would be C

mark me brainliest

Given that 1 inch = 2.54 cm, how many centimeters are there in 5 feet?

Answers

Answer: 152.4 cm

Step-by-step explanation:

Suppose ALI was told that relative to the other scores on a NTS test, his score was the 95th percentile i.e., his percentile score is 95. How do we interpret it?

Answers

We do a percentage 95% would be his score

!!!!!!!!!!!!!!!!!! help

Answers

Answer:

option 1

Step-by-step explanation:

It's given in the question that

a (first term of an A.P.) = 14

d (common difference between terms of A.P.) = -3

So,

2nd term will be = a + d = 14 + (-3) = 11

3rd term = a + 2d = 14 + 2×(-3) = 8

4th term = a + 3d = 14 + 3×(-3) = 5

All these terms are matching with option 1. So, option 1 is the correct option

HELP ME PLZ

A figure is shown in the image. What is the measure of the
highlighted angle?
А. 30
В. 60
С. 120
D. 65

Answers

Answer:

c is the correct answer i my guest

c is the answer to the question

Please answer i don't have muc time

Answers

I’m pretty sure it’s -25

Translate into a variable expression.
18 added to Z

Answers

Answer:

z + 18 or z+18

if you want i can even put it in word form.

Answer:

zjsjsjwjs1 balducci howbto

Step-by-step explanation:

hendrickson

True or False: Receipts are a way to document or keep track of your expenses.

Answers

Answer:

True

Step-by-step explanation:

I hope this is correct and have a great day

Answer:

True

Step-by-step explanation:

People save receipts to keep track of their previous purchases and expenses.

I hope this helps

what is the percent of change from 300,000 to 30,000 its 90​

Answers

Answer:

-90%

Step-by-step explanation:

8 pencils for 14 dollars

Answers

Answer:

1 pencil will be for 1.75 $

Answer:

$1.75 for each pencil.

Step-by-step explanation:

if you're asking the price for each pencil then its 1,75 for each because 8 multiplied by 1.75 is 14 :))

Axis of symmetry: x = -0.5; Vertex: (-0.5, 0.75); f(x) = x2 + x + 1
Axis of symmetry: x = -0.5; Vertex: (-0.5, -0.75); f(x) = x2 - x + 1
Axis of symmetry: x = -0.5; Vertex: (-0.5, 0.75); f(x) = -x2 + x
Axis of symmetry: x = -0.5; Vertex: (-0.5, 0.75); f(x) = x2 + 2x + 1

Answers

Graph of the question is missing so I have attached it

Answer:

Option A: Axis of symmetry: x = -0.5; Vertex: (-0.5, 0.75); f(x) = x2 + x + 1

Step-by-step explanation:

1) The axis of symmetry of quadratic graph is the vertical line that divides the graph curve into two congruent halves. In this case, it is: x = -0.5

2) Vertex is the point at which the graph curve changes direction or simply coordinates of the crest or trough of the curve.

The graph given has a trough with the coordinates: x = -0.5, y = 0.75. This is (-0.5, 0.75)

3) The roots of a quadratic equation are the points where the curve crosses the x-axis. In this case, it doesn't cross and so we have imaginary roots.

Now, formula for line of symmetry is; x = -b/2a

Thus; -b/2a = -0.5 or -b/2a = -1/2

Thus, b = 1 and a = 1

Our first and second terms will now be;

x² + x

Looking at the options, the only one with x² + x as it's first 2 terms is option A.

Thus, the complete equation will be x² + x + 1

Please help, solve a and b

Answers

Answer:

AE and BD use the same lines for both triangles so the angles for C are the same. this means that all the angles are congruent between both triangles.

Step-by-step explanation:

Answer:

The distance across the river d=175ft

Step-by-step explanation:

As◇ ABC& ◇ EDC are similar, BC/CD=AB/ DE

4x/7x=100/d

d=100×7/4=175

PLEASE HELP ME OUT FAST MULTIPLE CHOICE IF YOU DOMT KNOW PLEASE DONT RIGHT ANYTHING!

Answers

Answer:

B

Step-by-step explanation:

When you are finding the solution of two lines on a graph, you look at the intersection point and that is the answer. On the given graph, both lines intersect at (4, -3). Thus, that is the solution.

Best of Luck!

3. (03.01 LC) The domain of the following relation: R: {(-4, 8), (8, 10), (5, 4), (1, 6), (5, -9)} is (1 point) {8, 10, 4, 6, -9) {-4, 1, 5, 8) O {-4, 8, 5, 1, 5} No domain exists​

Answers

Answer:

{-4, 1, 5, 8}

Step-by-step explanation:

The domain of a relation are all of the inputs (or x-values) of it.

Identify the x-values shown.

{(-4, 8), (8, 10), (5, 4), (1, 6), (5, -9)}

{-4, 1, 5, 8} is the domain of the relation.

Hope this helps.

83, 15, 50, 28, 52, 96 - What is the mean? What is the mode? What is the median?

Answers

Answer: mean-54     mode-1     median-50

Step-by-step explanation:

Simplify this problem :

2 5/8 - 1 1/3

Answers

Answer:

-1 1/24

Step-by-step explanation:

Answer:

31/24 is the simplified answer

Step-by-step explanation:

ILL GIVE BRAINLEST ​

Answers

Answer:

the 4th one

Step-by-step explanation:

Answer:

147/2

Step-by-step explanation:

Hi! y=k/x

y=21/4 and x=14 so k=147/2

Please mark brainliest and let me know if you have any more questions!

A well known pharmaceutical manufacturer is manufacturing a newly developed vaccine, and is concerned about variability in the immune response of the recipients. They claim the variance in the immune response is 1.9. It is known that the immune responses follow a normal distribution. A random sample of 30 individuals were given the vaccine and their immune response recorded. It was found that they had a sample variance of 0.9. At the 5% significance level, test the hypothesis that the population variance of the immune response is equal to 1.9 against the two-sided alternative, that it is not. a. State the null and alternative hypothesis. b. Perform the hypothesis test. (You may use any of the methods we discussed that are appropriate for the problem.) c. State your decision. d. State your conclusion in full sentences.

Answers

Answer:

a

The  null hypothesis is [tex]H_o : \ \sigma^2 = 1.9[/tex]

The alternative hypothesis is  [tex]H_a : \ \sigma^2 \ne 1.9[/tex]  

b

 [tex]X^2 = 13.74[/tex]

c

The decision rule is  

Reject the null hypothesis

d

The conclusion is  

There is no sufficient evidence to conclude that the variance of the immune response is equal to 1.9.

Step-by-step explanation:

From the question we are are told that

   The variance is [tex]\sigma ^2 = 1.9[/tex]

    The sample size is  n =  30  

    The  sample variance  is  [tex]s^2 = 0.9[/tex]

    The level of significance is  [tex]\alpha = 0.05[/tex]

   

The  null hypothesis is [tex]H_o : \ \sigma^2 = 1.9[/tex]

The alternative hypothesis is  [tex]H_a : \ \sigma^2 \ne 1.9[/tex]  

Generally the test statistics  is mathematically represented as

          [tex]X^2 = \frac{ (n- 1 ) * s^2 }{ \sigma^2 }[/tex]

=>       [tex]X^2 = \frac{ (30 - 1 ) * 0.9 }{1.9 }[/tex]

=>       [tex]X^2 = 13.74[/tex]

Generally  from the degree of freedom is mathematically represented as

         [tex]df = n- 1[/tex]

=>      [tex]df = 30 - 1[/tex]

=>      [tex]df = 29[/tex]

Generally from the chi distribution table the critical value of  [tex]\frac{\alpha}{2} \ and \ 1 - \frac{ \alpha }{2}[/tex] at a degree of freedom  of  [tex]df = 29[/tex] is  

       [tex]X^2 _{ \frac{\alpha }{2} , df } = X^2 _{ \frac{0.05 }{2} , 29 } = 45.7[/tex]

=>     [tex]X^2 _{ 1 - \frac{\alpha }{2} , df } = X^2 _{1 - \frac{0.05 }{2} , 29 } = 16.0 5[/tex]

Gnerally from the values obtained we see that

         [tex]X^2 < X^2 _{ 1 - \frac{\alpha }{2} , df }[/tex] hence

The decision rule is  

Reject the null hypothesis

The conclusion is  

There is no sufficient evidence to conclude that the variance of the immune response is equal to 1.9.

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