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a  = log₁₀ 3

b = log₁₀ 7

By the  change- of- base theorem we have that   

a = log 3 / log 10   =   log 3

b = log 7 / log 10 =   log 7                   [Note that log 10  = 1 ]

Also note that 

2a  =  2 log₁₀3   = log₁₀3^2   =  log₁₀ 9  = log 9 / log 10   =  log  9

So

log ₇ 9  =   log  9  / log 7   =  2a / b 

Hi wiseowl!

Connect the centers like in the following diagram: https://imgur.com/WoRzy2n

This forms a right triangle with equal legs of 2 units. 

The hypotenuse is equal to the diameter of the smaller circle plus 2. 

We can right the equation: \(2^2+ 2^2=(2+2x)^2\\ 8=4+8x+4x^2\\ x^2+2x-1=0\) 

Solving for x, we get x = √2 - 1

I hope this helped,

Gavin

Find the centers of each of the 4  larger circles and connect them

This will form a square of 2 units on each side

And the distance from the center  of the circle at the top left to the  center of the circle on the bottom right  =

2√2  units

So......the diameter of the small circle =  2√2 - 2   = 2 [ √2 - 1]

And its radius is 1/2 of this  = [ √2 - 1 ]  units 

q(q + 1) (q + 2)(q + 3)  +  1   =

(q^2 + q)(q^2 + 5q + 6)  + 1   =

q^4  + 5q^3  + 6q^2

        +  q^3  + 5q^2 + 6q  + 1

_________________________

q^4  +  6q^3  + 11q^2 + 6q +  1

q^4  + 6q^3  + 9q^2 + 2q^2  + 6q +  1  =

(q^2 +3q)^2  + 2(q^2 + 3q)  +  1    =

([ q^2 + 3q ] + 1) ([ q^2  + 3q ]  + 1 )  =

(q^2  + 3q + 1)^2

Hello wiseowl! 

The diameter must pass through the midpoint of the circle. Furthermore, the midpoint of the diameter is the center of the circle. 

Therefore, from the diameter, we can find the radius length and center, finding the equation in the end. 

The midpoint of (-1,-3) and (5,1) is \((\frac{-1+5}{2},\frac{-3+1}{2})\) or \((2,-1)\).

The diameter length is \(\sqrt{(5-(-1))^2+(1-(-3))^2}=2\sqrt{13}\)

Then the radius must be \(\sqrt{13}\)

The equation of the circle must be \((x-2)^2+(y+1)^2=13\)

I hope this helped, 

Gavin

The only solution is x = 1.

q(q + 1)(q + 2)(q + 3) + 1 = q^4 + 2q^3 + 3q^2 + q + 1 = (q^2 + 3q + 1)^2

How many positive integers n are there such that n^2 + 96 is a perfect square?

n=2, 5, 10, 23

Oh, I double-counted!  I need to divide that by 2, so the answer is 864/2 = 432.

We need the radius to come back to the origin.  The smallest M that makes r = 0 is 2*pi/3.

There are 12 digaonals that have a length of P_1 P_2, which from the Sine Law, is sin (15 degrees).  There are 12 diagonals that have a legnth of P_1 P_3, which from the Sine Law, is sin (30 degrees).  We can appy the same reasoning to the other diagonals, which gives us a total sum of

(12 sin 15)^2 + (12 sin 30)^2 + (12 sin 45)^2 + (12 sin 60)^2 + (12 sin 75)^2 + (12 sin 90)^2 + (12 sin 105)^2 + (12 sin 120)^2 + (12 sin 135)^2 + (12 sin 150)^2 + (12 sin 175)^2 = 864.

Simplify logbase2 (8/sq rt 32).

Log(2, 8 / (4 * sqrt(2))

log(2,  2/sqrt(2))

Log(2, sqrt(2))

Log(sqrt(2)) / log(2)

= 1 / 2

3P + 7Q + 15R = 102. What is Q?

P = 2, Q = 3, R = 5

3*2 + 7*3 + 5*15 =102

thx so much

792 = 2^3 * 3^2 * 11. From factoring 792, it is quite obvious that: 2 x 11 =22 is the smallest positive integer, because: 792 x 2 x 11 =17,424 = 2^4 * 3^2 * 11^2 =which is the perfect square of 132.

75 larv / 5000 ml = .0015 l/ml

.0015 larv / ml  x 200 ml = 3 larv  should be in the sample

  out of 200 l arv 3

 3/200

Here is all your sequence factored. You decide what the answer is !!.

1 - 37 = 37
2 - 3737 =37  101
3 - 373737  = 3  7  13  37  37 
4 - 37373737  = 37  73  101  137  
5 - 3737373737  = 37  41  271  9091 
6 - 373737373737  = 3  7  13  37  37  101  9901
7 - 37373737373737  = 37  239  4649  909091  
8 - 3737373737373737  = 17  37  73  101  137  5882353  
9 - 373737373737373737  = 3  3  7  13  19  37  37  52579  333667  
10 - 37373737373737373737  = 37  41  101  271  3541  9091  27961  
  

only if the coeffficients are real 

Hi  Gore,

You have totally ignored the word uniformly which is probably fair enough since it made absolutely no sense.

Your logic is good but I did need to think about this red bit. 

I suppose it does makes sense.  I've just do not remember logic like that used before.

The probability of this occurrence is (pi-theta)/2pi, since F is equally likely to go anywhere on the circle.

Since the average value of theta between 0 and pi/2 is pi/4 , it follows that the overall probability for  is (1/2) - [(pi/4)/2pi]  or 3/8.

It is ok. I can see that it is logical.

Thanks very much for your very clear explanation.

Attn: whoisjo

Now you can change the question to a whole new one or delete it altogether.  Like you have done before.

You cheating little drop kick.

Isn't it just   4 + 5i ?  Complex roots come in pairs like   g +- k i

After a few calulations I find the other root to be

-6-3i

The sequence looks like this:
1, 4, 13, 40, 121,.......etc.
The closed form of the sequence is:
a(n) = 1/2 (3^n - 1) 

a(1978) = 1/2 * (3^1978 - 1)

Thanks, dude  !!!!