[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.
Learn more about logarithm here:
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the equation of a circle whose center is (-5, 3) and whose radius is 6?
Answer:
Center:(-5,3)
The center of a circle is a point from which all points on a circle are the same distance.
Radius:6
The radius of a circle is the length of a line segment from its center to its perimeter.
The radius is typically denoted as "r" or "R".
Diameter:12
The diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle.
Circumference or (or Permieter) = 2*π*R = 2*3.14*6 = 37.6991118430775
The circumference of a circle is the distance around it.
Area:113.097335529233
Area of a Circle is the amount of space occupied by the circle.The area of a circle is p times the radius squared, which is written: A = π*R2.
What is the lim as x approaches pi [integral 1+tan t from pi to x]/(pi sin x)
Apply l'Hopital's rule:
[tex]\displaystyle\lim_{x\to\pi}\frac{\displaystyle\int_\pi^x (1+\tan(t))\,\mathrm dt}{\pi\sin(x)}=\lim_{x\to\pi}\frac{1+\tan(x)}{\pi\cos(x)}=\frac{1+\tan(\pi)}{\pi\cos(\pi)}=\boxed{-\frac1\pi}[/tex]
where
[tex]\displaystyle\frac{\mathrm d}{\mathrm dx}\left[\int_\pi^x(1+\tan(t))\,\mathrm dt\right]=1+\tan(x)[/tex]
follows from the fundamental theorem of calculus.
p/2 = 55
(P divided by 2)
Please help
Answer:
p = 110
Step-by-step explanation:
p = 55 × 2
55 × 2 = 110
hope this helps
Answer:
p = 27.5
Step-by-step explanation:
p/2 = 55
p = 55/2
p = 27.5
Please mark me brainlest
help due in 30 min!!!!!!!!!!!!!!!
Answer:
They have the same slope
Step-by-step explanation:
slope=4
What is a factor that is less than 1?
Answer:
none
Step-by-step explanation:
Please help Im confused
Answer:
x=40
Step-by-step explanation:
to get from 3 to 15, you multiply by 5
multiply 8 by 5 and you get 40
x is 40
please give me brainliest!!<3
Someone please help!!
Answer:
A
Step-by-step explanation:
Hi! I think the answer is gonna be B! I could be wrong but that’s what I worked it out to! Good luck on the edge thing!!
please help, thanks :)
Answer:
It's C :)
Step-by-step explanation:
Next time check the y-intercept
Green has a intercept of 1
and purple has -3
x-2y what is the coefficient of x
Answer:
the coefficient of x is an implied 1
any variable that's by itself without a coefficient has an implied 1
Step-by-step explanation:
I hope this helps :)
All of the following would make a formal proof incomplete except what?
A) A lack of justifications
B) Incomplete sentences
C) Errors in logic
D) Incorrect reasons for statements
Urgent I need this for tonight please!!!
Answer:
b incorrect sentences
Step-by-step explanation:
got it right on my test
which of the following are functions.
Answer:
Graph ii and iii
Step-by-step explanation:
In graph ii and iii, every input has one and only output. Another way to check is by doing the vertical line test.
PLEASE ANSWER ASAP FOR BRAINLEST!!!!!!!!!!!!!!!!!!!!
The second angle of a triangle is three times as large as the first angle. The third angle is 40° greater than the first angle. How large are the angles?
The hypotenuse of a 45 degrees - 45 degrees - 90 degrees triangle is 5sqrt(6) Find the length of one leg of the right triangle
Answer:
Step-by-step explanation:
let the length of a leg of right angled associate triangle=x
x²+x²=(5√6)²
2x²=150
x²=150/2=75
x=√75=5√3
The table below represents Mark's assets and liabilities.
Item
Value
Pocket Money
$18
Checking Account Balance $45
Loan from friend
$14
Check for mowing lawn $25
What is the total of Mark's liabilities?
O $39
O $25
O $18
O $14
Answer:
18
Step-by-step explanation:
You took a math test with 20 questions on it. Your teacher said you got 90% of
the problems correct. How many questions did you get correct on the test?
a.
9
b.
15
C.
18
d.
90
Please hurry
Answer:
Hello as fast as i can answer 18.
Step-by-step explanation:
Reason 20 times 5 equals a 100
so 18 times 5 equals 90.
there for 18 is your answer.
-
Have a good day.
Answer:
C. 18
Step-by-step explanation:
90% of 20 is 18! Good luck!
What are the coefficients of this expression? x + y + 2
Answer:
1
Step-by-step explanation:
Since the variables dont have any numbers next to them, 1 is the coefficient for x and y
Please help I might flunk
8x = 88
I can’t figure this out 10 points if you answer
Answer:
divide 88 by the number of x
88/8= 11
X = 11
Answer:
x=11
Step-by-step explanation:
8x=88
You use inverse operation which means you would divide 8 on both sides to isolate x on one side resulting in x=11
Which of the following systems of linear inequalities is represented by the
solution graphed below?
Answer:
look it up
Step-by-step explanation:
why do you think that slowing down is sometimes called deceleration instead of acceleration?
Answer:
The "de" part means oppsite to whats behind it I think
Step-by-step explanation:
Answer:
Because acceleration means to gradually speed up, while deceleration means to gradually slow down.
When you plugged in your phone you had 36% left on your battery. It took 25 minutes to rwach 100% charge. What is the constant rate of change of % charge per minute?
Answer:
0.390%
Step-by-step explanation:
100-36=64
25÷64=0.390625
Answer:
2.6
explanation:
100-36=64
64/25= 2.6
Where do the lines y = 3x + 5 and y = -2x + 10 intersect?
9514 1404 393
Answer:
(x, y) = (1, 8)
Step-by-step explanation:
A graphing calculator quickly tells you the point of intersection.
__
You can equate the y-values and solve for x.
3x +5 = -2x +10
5x = 5 . . . . . . . . . add 2x-5
x = 1 . . . . . . . divide by 5
3·1 +5 = y = 8 . . . . substitute for x in the first equation
The point of intersection is (x, y) = (1, 8).
slope of the line (-3,-6) and (4,1)
Answer:
- 5 / 7
Step-by-step explanation:
1 - 6 = - 5
4 - - 3 = 7
Pls pls pls help ,,,,,,,,
Answer:
72 (d)
Step-by-step explanation:
I did the math
Find the roots of h(t) = (139kt)^2 − 69t + 80
the smaller root is:
the larger root is:
The answers will consist of algebraic expressions containing the parameter k.
What positive value of k will result in exactly one real root?
K = ?
Answer:
The positive value of [tex]k[/tex] will result in exactly one real root is approximately 0.028.
Step-by-step explanation:
Let [tex]h(t) = 19321\cdot k^{2}\cdot t^{2}-69\cdot t +80[/tex], roots are those values of [tex]t[/tex] so that [tex]h(t) = 0[/tex]. That is:
[tex]19321\cdot k^{2}\cdot t^{2}-69\cdot t + 80=0[/tex] (1)
Roots are determined analytically by the Quadratic Formula:
[tex]t = \frac{69\pm \sqrt{4761-6182720\cdot k^{2} }}{38642}[/tex]
[tex]t = \frac{69}{38642} \pm \sqrt{\frac{4761}{1493204164}-\frac{80\cdot k^{2}}{19321} }[/tex]
The smaller root is [tex]t = \frac{69}{38642} - \sqrt{\frac{4761}{1493204164}-\frac{80\cdot k^{2}}{19321} }[/tex], and the larger root is [tex]t = \frac{69}{38642} + \sqrt{\frac{4761}{1493204164}-\frac{80\cdot k^{2}}{19321} }[/tex].
[tex]h(t) = 19321\cdot k^{2}\cdot t^{2}-69\cdot t +80[/tex] has one real root when [tex]\frac{4761}{1493204164}-\frac{80\cdot k^{2}}{19321} = 0[/tex]. Then, we solve the discriminant for [tex]k[/tex]:
[tex]\frac{80\cdot k^{2}}{19321} = \frac{4761}{1493204164}[/tex]
[tex]k \approx \pm 0.028[/tex]
The positive value of [tex]k[/tex] will result in exactly one real root is approximately 0.028.
Write an equation for line L in point-slope form and slope-intercept form.
L is perpendicular to y=4x.
Answer:
[tex]y = -\frac{1}{4}x + c[/tex] --- Slope intercept
[tex]y - y_1 = -\frac{1}{4}(x - x_1)[/tex] --- Point slope
Step-by-step explanation:
Given
Perpendicular to [tex]y = 4x[/tex]
Required
Determine its equation
The general form of an equation is:
[tex]y = mx + c[/tex]
Where
[tex]m = slope[/tex]
By comparison:
[tex]m = 4[/tex]
Next, we calculate the slope of L.
Because L is perpendicular to [tex]y = 4x[/tex], the relationship between their slopes is:
[tex]m_2 = -\frac{1}{m}[/tex]
Substitute 4 for m
[tex]m_2 = -\frac{1}{4}[/tex]
The equation of L in slope intercept form is: [tex]y = mx + c[/tex]
Substitute -1/4 for m
[tex]y = -\frac{1}{4}x + c[/tex]
In point slope form, we have:
[tex]y - y_1 = m(x - x_1)[/tex]
Substitute -1/4 for m
[tex]y - y_1 = -\frac{1}{4}(x - x_1)[/tex]
Answer:
y-4=-1/4(x-1)
y=-1/4x+17/4
Step-by-step explanation:
fine the slope of the table below
out of 800 cards in a album each page of the album holds 8 cards if 66% of the album is empty how many pages are filled with cards
Answer:
ummm you are going to do terrible on on your test jk
Step-by-step explanation:
Which quadratic equation has roots -7 and 3/2 ?
Answer:
If the discriminant of the quadratic equation is zero then it will have 2 equal roots.
If the discriminant of the quadratic equation is greater than zero then it will have 2 different roots.
If the discriminant of the quadratic equation is less than zero then it will have no roots.
Step-by-step explanation:
hope this helped