If a = 5^x, b = 5^y and a^y×b^x = 25 then prove that: xy =1​

[tex]a = {5}^{x} \\ b = {5}^{y} \\ {a}^{y} \times {b}^{y} = 25 \\ prove \: ( {5}^{x} )^{y} \times ( {5}^{y} ) ^{x} = 25 \\ {5}^{xy} \times {5}^{yx} = {5}^{2} \\ {5 }^{xy + yx} = 5^{2} \\ 5^{2xy} = {5}^{2} \\ is \: \: 2xy = 2 \\ xy = \frac{2}{2} \\ xy = 1[/tex]

The prove that  xy is equal to 1 is given below.​

When you raise a number with an exponent, there comes a result.

Lets say you get

a^b = c

Then, you can write 'b' in terms of 'a' and 'c' using logarithm as follows

[tex]log_a(c) b = \log_a(c)[/tex]

'a' is called base of this log function. We say that 'b' is the logarithm of 'c' to base 'a'

Some properties of logarithm are:

[tex]\log_a(b) = \log_a(c) \implies b = c\\\\\log_a(b) + \log_a(c) = \log_a(b \times c)\\\\\log_a(b) - \log_a(c) = \log_a(\frac{b}{c})\\\\\log_a(b^c) = c \times \log_a(b)\\\\\log_b(b) = 1\\\\\log_a(b) \times log_b(c) = \log_a(c)[/tex]

Log with base e = 2.71828... is written as

ln(x)\ln(x)

simply.

Log with base 10 is written as

[tex]\log(x)[/tex]

We are given that;

a = 5^x, b = 5^y and a^y×b^x = 25.

Now,

To prove that xy = 1, we can use the properties of exponents and logarithms. Here are the steps:

1. Given that a = 5^x and b = 5^y, we can substitute these expressions into the equation a^y×b^x = 25 and get:

(5^x)^y × (5^y)^x = 25

2. Using the property that (a^b)^c = a^(bc), we can simplify the left-hand side of the equation and get:

5^(xy + yx) = 25

3. Using the property that a^b = c is equivalent to log_a(c) = b, we can take the logarithm of both sides of the equation with base 5 and get:

log_5(5^(xy + yx)) = log_5(25)

4. Using the property that log_a(a^b) = b, we can simplify the left-hand side of the equation and get:

xy + yx = log_5(25)

5. Using the property that log_a(b) = log_c(b) / log_c(a), we can change the base of the logarithm on the right-hand side of the equation to any other base, such as 10, and get:

xy + yx = log_10(25) / log_10(5)

6. Using a calculator or a table of logarithms, we can evaluate the right-hand side of the equation and get:

xy + yx ≈ 2 / 0.69897

7. Simplifying the fraction on the right-hand side of the equation, we get:

xy + yx ≈ 2.8614

8. Since xy and yx are symmetric, we can divide both sides of the equation by 2 and get:

(xy + yx) / 2 ≈ 2.8614 / 2

9. Simplifying both sides of the equation, we get:

xy ≈ 1.4307

10. Since x and y are integers, we can conclude that xy must be an integer as well. The only integer that is close to 1.4307 is 1.

Therefore, by logarithm properties we can say that xy = 1.

Learn more about logarithm here:

https://brainly.com/question/20835449

#SPJ2

Help me please......

Please help me ASAP I’ll mark Brainly

Please help! I’ll mark you as brainliest if correct

A square field has a perimeter if 3.6km what is the length of one side in metres​

Scenario: Multiplying Polynomials

Instructions:

View the video found on page 1 of this Journal activity.
Using the information provided in the video, answer the questions below.
Show your work for all calculations.
The Students' Conjectures

Emily and Zach have two different polynomials to multiply:

Polynomial product A:
(4x2 – 4x)(x2 – 4) Polynomial product B:
(x2 + x – 2)(4x2 – 8x)
They are trying to determine if the products of the two polynomials are the same. But they disagree about how to solve this problem.

1. Complete the table to summarize each student's conjecture about how to solve the problem. (2 points: 1 point for each row of the chart)

Classmate Conjecture
Emily



Zach



Evaluating the Conjectures

Who do you think is correct? (1 point)




Analyzing the Data

Table Method:

2. Use the table to find the products of the two polynomials. Write your answer in descending order. (4 points: 2 points for each product)

A) (4x2 – 4x)(x2 – 4)

4x2 –4x
x2
–4










B) (x2 + x – 2)(4x2 – 8x)

there's another part on my profile, can u do it to please!!

Write in exponential form, then evaluate. (5)(5)(5)(5)(5)​

8 + 3 × (4 ÷ 2) − 7



Step 1: 8 + 3 × 2 − 7

Step 2: 11 × 2 − 7

Step 3: 22 − 7

Step 4: 15



In which step does a mistake first appear?

Can someone turn this into an algebraic equation?
A community theater sold a total of 400 full-price tickets for adults and children. The price was $8.00 per adult ticket and $5.00 per child ticket. If the total
revenue was $2,750, how many adult tickets and how many children tickets were sold.

A 145 ft. building casts a 667 ft. shadow from the sun. To the nearest tenth of a degree, what is the angle of depression? Need it asap pls ;-;

A principal of $2600 is invested at 3% Interest, compounded annually. How much will the investment be worth after 14 years

can someone help me please look at picture

(1- 7n) • 5 simplify thank you

Is every relation also a function? Use the drop-down menus to show and explain your answer.
1. No or Yes
2. Relation or Function
3. Relation or Function
4. Relation or Function

PLEASE HELP ASAP! WILL GIVE BRAINLIEST!

6
If y varies directly as x and y = -7 when x = -28, find y when x = 20.
PLZ help ASAP

PLZ NEED HELP QUICK PLZ

Please help! I’ll venmo/ cash app $3 if you help or get right

Use the given cost table for the same product from two different companies to create a linear syst
Then solve the system to determine when the cost of the product will be the same and what the
be.
Two online retailers sell organic vanilla extract by the ounce using the following pricing chart.
Vanilla Extract (oz)
Chef Mate
c)
Grocery Gourmet
gla)
2
$13.50
$24.00
3
$18.25
$28.00
4
$23.00
$32.00
5
$27.75
$36.00
gin)
Both Chef Mate and Grocery Gourmet charges
for
ounces of vanilla extract.

Look in the document

Find three consecutive odd integers such that the sum of the smallest and twice the second is 14 more than the third.

Show your work

the sum for 3 times x and eleven

Solve 2x + 6 < 10 or 2x + 8 > 20. A. x < 2 and 2 > 2 B. I 2 O C. x < 1 or x > 6 O D. x < 2 or x > 6​

What is the result of subtracting X^2-2x+3 from 3x^2+x-2

How many more pound of granola than banana chips does she buy? Item: Pretzels:7/8,Peanuts: 4/8, Raisins: 2/8,Banana chips: 3/8,Granola: 5/8​

help please im bad at math ​

A breakfast diner ordered milk with 1.5% fat but 30 liters of milk with 5% fat were delivered. How many liters of milk with 0% fat should the waitress add to the delivery to get milk with 1.5% fat?

I will give brainiest to whoever answers correctly !!

Hey did I do this right?

Can u guys help me quick (write an expression)

2x – 3y = -5
4x - 4y = -4
Is (2.3) a solution of the system?