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.
What is logarithm and some of its useful properties?
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.
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the sum for 3 times x and eleven
Answer:
3x +11 = 32
Step-by-step explanation:
hope this is correct! <3
Answer:
3x +11 = 32
3x = 32-11
3x = 21
x = 21/3
x = 7
check answer:
3(7)+11 = 32
21 + 11 = 32
32 = 32
LHS = RHS
Hence, verified
What is the slope of the line which passes through (1, 4) and (0, 1)?
Determine the above sequence converges or diverges. If the sequence converges determine its limit
Answer:
This series is convergent. The partial sums of this series converge to [tex]\displaystyle \frac{2}{3}[/tex].
Step-by-step explanation:
The [tex]n[/tex]th partial sum of a series is the sum of its first [tex]n\!\![/tex] terms. In symbols, if [tex]a_n[/tex] denote the [tex]n\![/tex]th term of the original series, the [tex]\! n[/tex]th partial sum of this series would be:
[tex]\begin{aligned} S_n &= \sum\limits_{k = 1}^{n} a_k \\ &= a_1 + a_2 + \cdots + a_{k}\end{aligned}[/tex].
A series is convergent if the limit of its partial sums, [tex]\displaystyle \lim\limits_{n \to \infty} S_{n}[/tex], exists (should be a finite number.)
In this question, the [tex]n[/tex]th term of this original series is:
[tex]\displaystyle a_{n} = \frac{{(-1)}^{n+1}}{{2}^{n}}[/tex].
The first thing to notice is the [tex]{(-1)}^{n+1}[/tex] in the expression for the [tex]n[/tex]th term of this series. Because of this expression, signs of consecutive terms of this series would alternate between positive and negative. This series is considered an alternating series.
One useful property of alternating series is that it would be relatively easy to find out if the series is convergent (in other words, whether [tex]\displaystyle \lim\limits_{n \to \infty} S_{n}[/tex] exists.)
If [tex]\lbrace a_n \rbrace[/tex] is an alternating series (signs of consecutive terms alternate,) it would be convergent (that is: the partial sum limit [tex]\displaystyle \lim\limits_{n \to \infty} S_{n}[/tex] exists) as long as [tex]\lim\limits_{n \to \infty} |a_{n}| = 0[/tex].
For the alternating series in this question, indeed:
[tex]\begin{aligned}\lim\limits_{n \to \infty} |a_n| &= \lim\limits_{n \to \infty} \left|\frac{{(-1)}^{n+1}}{{2}^{n}}\right| = \lim\limits_{n \to \infty} {\left(\frac{1}{2}\right)}^{n} =0\end{aligned}[/tex].
Therefore, this series is indeed convergent. However, this conclusion doesn't give the exact value of [tex]\displaystyle \lim\limits_{n \to \infty} S_{n}[/tex]. The exact value of that limit needs to be found in other ways.
Notice that [tex]\lbrace a_n \rbrace[/tex] is a geometric series with the first term is [tex]a_0 = (-1)[/tex] while the common ratio is [tex]r = (- 1/ 2)[/tex]. Apply the formula for the sum of geometric series to find an expression for [tex]S_n[/tex]:
[tex]\begin{aligned}S_n &= \frac{a_0 \cdot \left(1 - r^{n}\right)}{1 - r} \\ &= \frac{\displaystyle (-1) \cdot \left(1 - {(-1 / 2)}^{n}\right)}{1 - (-1/2)} \\ &= \frac{-1 + {(-1 / 2)}^{n}}{3/2} = -\frac{2}{3} + \frac{2}{3} \cdot {\left(-\frac{1}{2}\right)}^{n}\end{aligned}[/tex].
Evaluate the limit [tex]\displaystyle \lim\limits_{n \to \infty} S_{n}[/tex]:
[tex]\begin{aligned} \lim\limits_{n \to \infty} S_{n} &= \lim\limits_{n \to \infty} \left(-\frac{2}{3} + \frac{2}{3} \cdot {\left(-\frac{1}{2}\right)}^{n}\right) \\ &= -\frac{2}{3} + \frac{2}{3} \cdot \underbrace{\lim\limits_{n \to \infty} \left[{\left(-\frac{1}{2}\right)}^{n} \right] }_{0}= -\frac{2}{3}\end{aligned}}_[/tex].
Therefore, the partial sum of this series converges to [tex]\displaystyle \left(- \frac{2}{3}\right)[/tex].
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?
Answer:
70 Litres
Step-by-step explanation:
Answer:
70 Liters
Step-by-step explanation:
Welcome
Drag each number to the correct location on the table. Classify the numbers according to their positions on a vertical number line. -4.5 4 -2.5 0.5 -0.5 1.5 Below -2 Between -2 and 1 Above 1
Answer:
Below -2 Is -4.5 - 2.5 then between -2 and 1 -0.5 and above 1 is 0.5, 1.5, and 4
Step-by-step explanation:
help please im bad at math
Answer:
32$
Step-by-step explanation:
There's a 25% discount, so,
After cutting discount, the total will be 9.78 -(9.78×25%)
=7.335
Again, after adding tax, total amount will be : 7.335+ ( 9.78×7.75%)
=8.09
So, the total cost of 4 DVD is 4× 8.09 =32.36
The length of a basketball court is (2x-6). The width of the court is (3x + 1). Write the expression for the perimeter of the basketball court.
Answer:
the expression of perimeter is 2(l+b)
PLEASE HELP ASAP! WILL GIVE BRAINLIEST!
Answer: g it is 2x+21 ansd h is 3x+24
Step-by-step explanation:
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.
Answer:
Step-by-step explanation:
400/8.00/5.00/2750 tell me if incorect
so I really need help. you have to set fractions equal to each other and try and find out how many prefer sweaters and whatnot. Also, the answer isn't supposed to be a fraction, it's supposed to be a whole number, you find the answer by using fractions with the information provided in the word problem above ^
I hope that makes sense. lol, thank you :)
Answer:
84
Step-by-step explanation:
52%of math problem are pointless
52%of math problem = pointless
52% of 175
(52/100)×175
91 =pointless
math problem - pointless =not pointless
175 - 91 = 84
Plz help me I am timed plz don’t guess
DeArah made 88, 90, and 92 on her math tests this grading
period. What is the mean of her math test scores?
OA) 89
OB) 90
OC) 91
OD) 92
Answer:
B. 90
Step-by-step explanation:
Add them all, divide by 3.
88 + 90 + 92 = ?
270 = 270
270/3 = 90
Answer: 90
Step-by-step explanation:
You just add up all the numbers then divide by how many numbers there are.
( 88 + 90 + 92 = 270 ) There are 3 numbers so take 270 / 3 = 90.
slope of the line for (1,2) and (2,-3)
Answer:
-5/1
Step-by-step explanation:
Remember: Rise over Run
To solve this...
You will need the equation:
y² - y¹
x² - x¹
So now you have to label ( 1 , 2 ) and ( 2 , - 3 ) as either x¹ or y¹ etc.
x¹ = 1 and y¹ = 2
x² = 2 and y² = -3
Now you know your plug ins. Just solve it.
Your equation should look like this now:
-3 - 2 -5
------- = ----- ( so this is your answer )
2 - 1 1
If the following figure is a rectangle, find wz.
Answer:
Step-by-step explanation:
as both sides are equal so
7x-6 = 3x+14
7x-3x = 14+6
4x = 20
x= 5
Now, put x in 7x-6 to find WZ
7(5)-6 = 29
Also, 3(5)+14 = 29
The measurement of the side WZ of rectangle WXYZ is 29.
What are rectangles?A rectangle is a closed 2-D shape, having 4 sides, 4 corners, and 4 right angles (90°). The opposite sides of a rectangle are equal and parallel.
Given that, WXYZ is a rectangle, we need to find WZ,
As we know that, a rectangle is a type of parallelogram and opposite sides of a parallelogram are equal and parallel,
WZ and XY are opposite sides of the rectangle, therefore, are equal.
Thus,
WZ = XY
7x - 6 = 3x + 14
4x = 20
x = 5
Put x =5 in 7x - 6
WZ = 7(5) - 6
WZ = 29
Hence, the measurement of the side WZ of rectangle WXYZ is 29.
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(1- 7n) • 5 simplify thank you
Answer:
5-35n
Step-by-step explanation:
5-35n
Just open the parenthesis and multiply by 5.
Answer:
5-35n
Step-by-step explanation:
Choose the answer that represents the expression
B4
b*b
b*b*b*b
b*b*b
Answer:
4 B
Step-by-step explanation:
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
$36.00
gin)
Both Chef Mate and Grocery Gourmet charges
for
ounces of vanilla extract.
Answer:
209 and 369 that's the answer
Calculating Percents 2
You may use a calculator for all calculations
Part 1 - Write each percent as a decimal equivalent
120%
2 15%
3 45%
4 50%
51%
63%
75%
8 6%
Answer: 120% is 1.2 in decimal form.
215% 2.15
345% 345
450% 4.50
51% 0.51
63% 0.63
75% 0.75
86$ 0.86
Step-by-step explanation:
Helppppp Me!!!!!!!!!
Answer:
Top right
Step-by-step explanation:
Answer:
The bottom right triangle is both acute and scalene.
Explanation:
A scalene triangle is a triangle in which all 3 sides have different lengths. An acute triangle has 3 different angles under 90º. This triangle has 3 different lengths (8.3ft, 9ft, and 10 ft) and angles less than 90º (51º, 58º, and 71º); therefore, it's both acute and scalene.
(Worth 50 points. Don't post your answer unless you have good ideas) Solve the following problem:
Which triangles are similar ?
Answer: ΔABC ≅ ΔADB
Step-by-step explanation:
In order to be similar, they must have the same angles.
ΔABC ΔADB ΔDBC
∠A = 180 - (2a + b) ∠A = 180 - (2a + b) ∠D = 180 - (a + b)
∠B = a + b ∠D = a + b ∠B = b
∠C = a ∠B = a ∠C = a
ΔABC and ΔADB have the same angles so they are similar.
Please help! I’ll venmo/ cash app $3 if you help or get right
A principal of $2600 is invested at 3% Interest, compounded annually. How much will the investment be worth after 14 years
Answer:
3692
Step-by-step explanation:
Multiply the starting amount, by the percent, and them multiply by how many years (2600*0.03%*14)
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!!
Answer: there's one more screen shot but the limit is 5... so ill type it out!
7. Are the two products the same when you multiply them vertically? (1 point)
Yes they are
Making a Decision:
8. Who was right, Emily or Zach? Are the products the same with the three different methods of multiplication? (1 point)
They were both correct
9. Which of these three methods is your preferred method for multiplying polynomials? Why? (1 point)
I prefer the first one, Table method. To me it is easier and the fastest out of them all.
if there is a trophy that is 13 inches tall and it's shadow is 26 inches and there's another trophy that's 7 inches tall, whats is it's height?
Please help! I’ll mark you as brainliest if correct
Answer:
Both answers are correct
please answer this asap giving brainliest!!!
Answer:
b = 39
Step-by-step explanation:
Opposite sides of a parallelogram are congruent to each other. This implies that:
ST = VU
Therefore:
2b - 39 = b (substitution)
2b - 39 - b = 0 (subtraction property of equality)
b - 39 = 0
b = 0 + 39 (addition property of equality)
b = 39
Help me please......
Answer:
i believe it would be -3
Step-by-step explanation:
9 + -12 = -3
Do this worksheet if someone do I will make him or her brainly if correct
Step-by-step explanation:
1) 15 % games lost
2) 3/20
3) 271/500
4) 12/25
Look in the document
Please help me ASAP I’ll mark Brainly