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Mr. BB, your presentation is SLOP. Anyone who can figure out the solution from your slop does not need the slop in the first place. 

\(\log_{4n} 40\sqrt{3} = \log_{3n} 45\\ \text{Antilog form }\\ 4^x n^x=40\sqrt{3}\\ 3^x n^x=45\\ \left(\dfrac{4}{3}\right)^x=\dfrac {1}{9}*{8\sqrt{3}} = \dfrac{8}{3\sqrt 3}\rightarrow x=\dfrac {3}{2}\\ 8n^{\frac {3}{2}}=40\sqrt 3\\ n= 3^{\frac13} * 5^{(\frac23)} \rightarrow n^3= 3 * 5^2 =75 \)

_{Tell me, Mr. BB, do you actually find diamonds in SLOP, or do you find a diamond and throw SLOP on it? (Like, I don't already know.)}

GA

The radius of the circle  is :     sqrt [ (100) / pi ]  =  10 / sqrt (pi)

We can find 1/2 of the length of the side of the triangle thusly :

[ (1/2)x ] /  sin (60)  =  [10/ sqrt (pi)]  / sin (30)

x / [ 2 sqrt (3) /2 ]  =  2 * 10 / sqrt(pi)

 x / sqrt (3)   =  20/ sqrt(pi)

x =  20 sqrt (3)  / sqrt (pi)   =   20  sqrt (3/pi) ≈ 19.54 cm

italy

log_4n 40sqrt3 = log_3n 45, solve for n

log(40sqrt(3))log(3n) =log(45)log(4n)

1.840620log(3n) = 1.653213log(3n) divide both sides by 1.6532131

1.11335996log(3n) =log(4n)

Solve for n:

(log(n) + log(4))/(log(n) + log(3)) = 1.11336

Simplify and substitute x = 1/(log(n) + log(3)).

(log(n) + log(4))/(log(n) + log(3)) = (log(4/3))/(log(3) + log(n)) + 1

= x log(4/3) + 1:

log(4/3) x + 1 = 1.11336

Subtract 1 from both sides:

log(4/3) x = 0.113359

Divide both sides by log(4/3):

x = 0.394043

Substitute back for x = 1/(log(n) + log(3)):

1/(log(n) + log(3)) = 0.394043

Take reciprocals of both sides:

log(n) + log(3) = 2.5378

Subtract log(3) from both sides:

log(n) = 1.43918

Cancel logarithms by taking exp of both sides:

 n = 4.21725, n^3 =~75

Yea me too. Although some of the gore I could have done without LOL.

2 log (a - 2b  )  = loga + log b

log (a - 2b)^2  = log (a*b)

This implies that

(a - 2b)^2  =  ab

a^2 - 4ab + 4b^2  = ab

a^2 - 5ab + 4b^2  = 0

(a - 4b) (a - b)  = 0

This implies that

a = 4b   ⇒   a/b  =  4      or

a - b  = 0  ⇒   a  = b  ⇒   a/b  =  1

(4/x+2)/(1/2x-8)   =   \(\frac{\frac4x+2}{\frac12x-8}\)

Only the first number after the division sign goes in the denominator when there aren't parenthesees.

1/2x   means   1/2 * x   which is   \(\frac12\) * x

and

1/(2x)   means   \(\frac1{2x}\) 

Well maybe next time then. I remember a few times you got into it with Nauseated. These were always entertaining...and sometimes gory.  I miss the good-old-days. 

A)  x =1.5

B) x=3

C) not sure but maybe x = -2

maybe misunderstood the question so ignore answers

AC / AB  =  3 / 4  ⇒  AB  = (4/3)AC

AB / BC  = 2/ 1   =  (1/2)AB = BC ⇒ AB = 2BC

AC / BC  = 7/4 ⇒  AC  = (7/4)BC

A. 

AC   = x

AB  = 2

AB  = (4/3)(AC)

2 = (4/3)x

6/4  = x  = 3/2

B.

AB  = 16x

CB  =  24

(1/2)AB  = CB  = BC

(1/2)AB  = 24

(1/2)16x  =  24

8x = 24

x  = 3

C. ABC  = 21

     BC = 2x + 4

This implies that

AB + AC + BC  =  21

Put everything in terms of BC

2BC + (7/4)BC + BC = 21        { sub  2x + 4  for BC }

2 (2x + 4)  + (7/4)(2x + 4)  + (2x + 4)  = 21

4x + 8  + (7/2)x + 7 + 2x + 4  = 21

(19/2)x + 19  = 21

(19/2)x  = 2

x  = (2/19) * 2   =  4/19

Proof :

BC   =  8/19 + 4  =  84/ 19  

AC  = (7/4)BC = (7/4)(84/19)  = 147/19  

AB = 2BC = 2(84/19)  =  168/19

(84 + 147 + 168)  / 19  =  21

2log(a-2b) = loga + logb  so log(a-2b)² = logab

therefore  (a-2b)² = ab

expanding : a² -4ab + 4b² = ab

ie              a²  - 5ab + 4b² = 0

divide both sides by b² : a²/b² -5ab/b² + 4 =0

ie              (a/b)² - 5(a/b) + 4 = 0

solving the quadratic for a/b gives (a/b - 1)(a/b - 4) = 0

therefore  a/b =1 or a/b = 4

Sorry to disappoint you   

question, sir, the original one is like this 4 over x plus 2 divided by 1 over 2x minus 8, so how can you separate out 1/2 and x because its like 1 divided 2x so im confused...

You are correct......I had already looked it up.  Thanx !  

y = men

x= women     (see what I did there?  Haha)

2x-3 = y

and  x+2 =y        so    2x-3=x + 2   (assuming there can be more than12 jurors)

  x = 5 women    y = 7 men     12 jurors total

(if you add two women, the number of men and women will be equal, but 14 jurors total)

Let  J  be the number of miles between Janet and the office.

Let  L  be the number of miles between Lynn and the office.

Janet and Lynn live 8 mi apart in opposite directions from their office.

J + L  =  8

Lynn lives 1 mi less than twice as far from the office as Janet does.

L  =  2J - 1    Use this value for  L  in the first equation.

J + (2J - 1)  =  8

3J - 1  =  8

3J  =  9

J  =  3     Use this value for  J  to find  L .

L  =  2J - 1

L  =  2(3) - 1

L  =  5

Janet lives  3  miles from the office, and Lynn lives  5  miles from the office.

Let  m = the number of men

Let  w = the number of women

There are 3 fewer men than twice the number of women.

m  =  2w - 3

If there were 2 more women on the jury, the numbers of men and women would be equal.

w + 2  =  m       Subtract  2  from both sides of this equation to solve for  w .

w  =  m - 2        Now we can use this value of  w  in the first equation.

m  =  2w - 3

m  =  2(m - 2) - 3

m  =  2m - 4 - 3

m  =  2m - 7

m - 2m  =  -7

-m  =  -7

m  =  7  =  the number of men

Here is the diagram I sketched:

The radius of the circle  is :     sqrt [ (100) / pi ]  =  10 / sqrt (pi)

We can find 1/2 of the length of the side of the triangle thusly :

[ (1/2)x ] /  sin (60)  =  [10/ sqrt (pi)]  / sin (30)

x / [ 2 sqrt (3) /2 ]  =  2 * 10 / sqrt(pi)

 x / sqrt (3)   =  20/ sqrt(pi)

x =  20 sqrt (3)  / sqrt (pi)   =   20  sqrt (3/pi) ≈ 19.54 cm

( 20 cm  rounded to 2 sig figs )

First find the radius of the circle   area = pi r^2   r = 5.64 cm

Then   tan 30 = 1/2 / (sqrt(3) /2)) = r/(1/2x)   where x is side length

1/2 / sqrt 3/ 2 =  r/(1/2 x)

1/2 x  * .57735 = r   = 5.64    x = 19.53 ~~~ 20 cm (two sig-figs  

 See diagram below:

Ooops ...forgot diagram (and I found an error...fixed)

It’s “half-lives” with a hyphen. CPhil used it, but most mathematicians don’t use this because they think it means to subtract. 

Part of the fun of figuring out the math problem is figuring out the written mush the answerer gives along with the solution.

Another take (but similar)

500 e^-kt  = 250       to solve for k     we will let t= number of half lives= 1    (as in ONE half life)

k = ln (250/500) / -1    k=.693147

Now for the sample

500 e^-.693147 t = 3.90625

t = #half lifes =   ln (3.90625/5000) / - .693147     t = 7.00 half lifes  (or is it half LIVES ?)

I got it never mind!

6xz√(2xz)

I am inperesed