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
+2440 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
http://www.pleacher.com/mp/probweek/p2003/a011303.html
https://web2.0calc.com/questions/andy
[plagiarized? Or was it conjured from "Set Theory"? As phony as a 3-dollar bill !. Utterly Shameless!]
+2440 Long ago, your ship sailed and sank. No one thought it worthy of salvage— then or now. It needs warning buoys because of the toxic waste that it leaches.
An arsonist thinks all fires start because of arson.
A thief thinks all missing items are stolen.
An embezzler thinks any missing funds are embezzled.
A plagiarist thinks every posted solution is plagiarized.
I wasn’t accusing you of plagiarism, Mr. BB. However, you didn’t cite the Mathematica program from which you copied your pasted SLOP. The slipper fits because it belongs to you. You are what you are, Mr. BB.
You are gifted Mr. BB, you are the only person I know that takes a brilliantly integrated mathematical program and finds a noxious use for it. Another example of throwing slop on a diamond.
GA
