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Take the log of base 2 of each side.

2x = log2 15

2x = 3.90...

x = 3.9/2 = 1.9534...

y  =  x² - 3x + 2

To find the roots (a,0) and (b,0), let y = 0:

y  =  x² - 3x + 2      --->     0  =  x² - 3x + 2      --->     0  = (x - 2)(x - 1)

                                                                       --->          x = 2  or  x = 1     --->     (2,0) and (1, 0)

The point where the graph crosses the y-axis has an x-value of 0:

y  =  0² - 3(0) + 2     --->     y = 2      --->     (0,2)

a = 1 and b = 2 and c = 2     or     a = 2 and b = 1 and c = 2

What exactly is 12 percent?

12 percent is 12/100. 

32 is a full number. So, it would equal 32/1.

"of" basically means multiply. To find 12 percent, we would multiply 12/100 by 32/1.

The answer is 3.84

Ok. So, you want to find n. To do that, we can first combine like terms. 6n+2n. Just like regular addition, 8n. The outcome would be 8n=72. So, if we want to find n, we need to somehow eliminate the 8. Since 8 multiplied by n would give us 72, we need to divide eight from both sides to get our final answer. So 72/8=9. n=9

Thank you for the correct way to answer my log question. I don't know what I would have done from this point. You have made this expience of learning possible for me to move on. Thanks again

105625

I looked on Google, but their numbers (100 billion stars, 1 billion trillion stars, 10 billion galaxies, 200 billion galaxies, 10^21 planets, 10^24 planets) were contradictory, so I used the numbers that made the most sense.

A typical galaxy has 100 billion stars. There are about 10 billion galaxies in the observable universe. So, there are 1,000,000,000,000,000,000,000 stars. A 2012 analysis put it at 1.6 planets per star. But, due to the fact that there is no universal time (only relative), the farther away something is, the farther back in time you observe it to be (if you were 65 million light-years from Earth, you would see Earth as it was 65 million years ago, when the dinosaurs went extinct). So, lets say that the average is about 1 planet per star (the above analysis was just for the Milky Way), so there are 1,000,000,000,000,000,000,000 stars, 1,000,000,000,000,000,000,000 planets, and 2,000,000,000,000,000,000,000 stars and planets in the universe, or 2.0*10^21 planets and stars in the universe (note that the information here is a little outdated, if you find better statistics, just swap out my numbers for yours and repeat my calculations).

By writing the function as  P(x)  =  (x - 3)² + 1,  you are showing that the minimum y-value of the function is 1.

[Since (x - 3)² has either a value of 0 or a positive value for any choice of x, the graph rises from the minimum value, which occurs when the value of (x - 3)² = 0. When the "+1" is added, the minimum y-value must be 1.]

Since the minimum y-value must be 1, choice 1) is incorrect because zeroes occur only when the y-value is 0.

Choice 2) is correct.

Choice 3) is incorrect because the minimum is 1, not 3.

Choice 4) is incorrect because there are no zeroes.

To find the largest power of 2 that divides into (2⁴)!:

First:  simplify (2⁴)!   --->   (2⁴)!  =  (16)!

16! = 16 x 15 x 14 x ... x 2 x 1

But, if we want to find a number that divides into 16!, we don't really care about the odd numbers in  16 x 15 x ... x 2 x 1.

All we need to look at are the even factors:  16 x 14 x 12 x 10 x 8 x 4 x 2.

16  = 2⁴

14 = 2 x 7

12 = 2² x 3

10 = 2 x 5

8 = 2³

6 = 2 x 3

4 = 2²

2 = 2

So, the total number of factors is 4 + 1 + 2 + 1 + 3 + 1 + 2 + 1  =  15     <---   Answer

As a check:  16! / 2¹⁵  =  638 512 875  (no further factors of 2)

Answer: Tony

Let the width  = w

Let the length = 2w + 4

So.....

l * w  = 48

Substituting, we have

(2w + 4) (w)  = 48       simplify

2w^2 + 4w   = 48       subtract 48 from both sides

2w^2 + 4w  - 48  =  0      divide through by 2

w^2 + 2w  -  24  =  0          factor

(w + 6) (w - 4)   = 0       set both factors to 0   and w = -6 feet [ not possible ]  or w = 4ft

And the length, l,    = 2w + 4  =  2(4) + 4  = 8 + 4   = 12 ft

Note the reason for not be able to take a negative log.......

If we had

log (-2)     ......this would imply that  there is some "x" such that

10^x  = -2

But notice that 10 raised to  any  "real" power is always positive....therefore.......there is no "real" power "x"   that we can raise 10 to that will result in a negative.......

At the end of the 6th year [= the beginning of the 7th ], there were 449 students

At the end of the 12th year [ = the beginning of the 13th] , there were 750 students

So we have these two equations where a₁ is the beginning enrollment [in the first year]   and d is the yearly [constant ] increase

449 = a₁  + d(6 -1)            →  449 = a₁ + 5d        (1)

750  = a₁ + d(13 - 1)         →  750  = a₁  + 12d     (2)

Subtract (1)  from (2)

301  = 7d        divide both sides by 7

43  = d    this is the yearly increase

And using (1)   we can find the beginning enrollment, a₁

449 = a₁  + 5(43)

449 = a₁ + 215    subtract 215 from each side

234 = a1

So.......the equation giving the number of students for any year,  n  =

a_n   =  234 + 43 (n - 1)

I'll write here the solution included in my book:

logx + log(x+1) = log2

logx(x+1)= log2

logx(x+1)-log2=0

and then

log(x(x+1)/2)=0  (can someone explain me this step plz)

x(x+1)/2= 10exp0=1  (I feel this is some basic log law but I can't remember what it is exactly, help haha)

Then they multiply each side by 2 to get rid of the /2:

x^2+x-2=0  (why do they end up with "x^2" and "-2" when they just get rid of the /2?)

then they say "logx(x+1)=log2" means that "x(x+1)=2" and that "x^2+x-2=0"

Then they find rationnal roots for "x^2+x-2=0" which are "x=1" and "x=-2" (that's fine for me)

Seems that "x=1" makes sense because "log1+ log(1+1)=log2"

but that "x=-2" because " log of a negative number is undefined" (can someone refresh me on that).

Thanks a lot guys!

logx+log(x+1)=log2    

Remember that if we have log a + log b, we can write this as log [a * b]........so....

log x + log (x + 1)   =  log [ x (x + 1) ]  =  log [x ^2 + x]

So

log [ x^2 + x  ]    = log 2         we can just forget the logs and solve this equation :

x^2 + x  = 2      subtract 2 from both sides

x^2 + x  - 2  = 0         factor

(x -1) (x + 2)   = 0       and setting each factor to 0, x = 1  or x = -2

Since we can't take the log of a negative number, x = - 2 is not a solution

Check  x = 1

log (1) + log (1 + 1) =  log 2 ?

0 + log 2   = log 2

log 2 = log 2       OK!!!

Solve for x:
log(x)+log(x+1) = log(2)

log(x)+log(x+1) = log(x (x+1)):
log(x (x+1)) = log(2)

Cancel logarithms by taking exp of both sides:
x (x+1) = 2

Expand out terms of the left hand side:
x^2+x = 2

Add 1/4 to both sides:
x^2+x+1/4 = 9/4

Write the left hand side as a square:
(x+1/2)^2 = 9/4

Take the square root of both sides:
x+1/2 = 3/2 or x+1/2 = -3/2

Subtract 1/2 from both sides:
x = 1 or x+1/2 = -3/2

Subtract 1/2 from both sides:
x = 1 or x = -2

log(x)+log(x+1) ⇒ log(-2)+log(1-2) = 2 i π+log(2) ≈ 0.693147+6.28319 i
log(2) ⇒ log(2) ≈ 0.693147:
So this solution is incorrect

log(x)+log(x+1) ⇒ log(1)+log(1+1) = log(2) ≈ 0.693147
log(2) ⇒ log(2) ≈ 0.693147:
So this solution is correct

The solution is:
Answer: |x = 1

Here is [roughly] Euclid's proof of this......

Let the greatest known prime = P

Now, multiply all the known primes together [including P ] and let this number  = M

Now....add 1 to M.......so.....the new number is M + 1

If we divide M + 1 by any of the known primes, we get a remainder .......so...........either M + 1 itself is prime, and therefore greater than the greatest prime, P,  or there exists a prime greater than any of the knowm primes which divides  M and which is also greater than P

Therefore.......the list of primes is infinite

If Maya pays 1 peso per hour for every hour in the 15-day period, then she would pay:

15 days x 24 hours =360 hours

360 x 1 =360 pesos in connection fees. This would be on top of 999 pesos, which her monthly access fee. So that her total cost would be:

999 + 360 =1,359 pesos.

Yes!! There are.

sqrt(4096) = 64 or -64

-sqrt (4096) = - (-64) = 64  ???   maybe?

1+2+3+4+5+.....= -0.083333.. This is the sum of Riemann's zeta function at s=-1

1^3+2^3+3^3+4^3+.....=ζ(3) = 1.20205690315959428539973..........etc.

This is called "Aprey's constant", which is the sum of Riemann's zeta function at s=3

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You have an unbalanced set of parentheses.

Enter:

Press equals to get:

.

probability = number that went to right field/total number

p = 20/67