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Sep 24, 2016
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Compare Melody's solution to this step-by-step solution by Wolfram/Alpha!. I think Melody's solution is more elegant than W/A!!. So winded!!.

 

Solve for N over the real numbers:
-10000+40396.2 1.07^(-N)-21428.6 (1-1.07^(-N)) = 0

-10000+40396.2 1.07^(-N)-21428.6 (1-1.07^(-N)) = -10000+201981 (4/107)^N 5^(2 N-1)-150000/7 (1-(100/107)^N):
-10000+201981 (4/107)^N 5^(2 N-1)-150000/7 (1-(100/107)^N) = 0

Factor 107^(-N) and constant terms from the left hand side:
1/35 107^(-N) (1413867 4^N 5^(2 N)+3 4^(N+2) 5^(2 N+6)-1100000 107^N) = 0

Multiply both sides by 35:
107^(-N) (1413867 4^N 5^(2 N)+3 4^(N+2) 5^(2 N+6)-1100000 107^N) = 0

Split into two equations:
107^(-N) = 0 or 1413867 4^N 5^(2 N)+3 4^(N+2) 5^(2 N+6)-1100000 107^N = 0

107^(-N) = 0 has no solution since for all z element R, 107^z>0:
1413867 4^N 5^(2 N)+3 4^(N+2) 5^(2 N+6)-1100000 107^N = 0

4^N 5^(2 N) = e^(log(4^N)) e^(log(5^(2 N))) = e^(N log(4)) e^((2 N) log(5)) = e^(2 N log(5)+N log(4)) and 4^(N+2) 5^(2 N+6) = e^(log(4^(N+2))) e^(log(5^(2 N+6))) = e^((N+2) log(4)) e^((2 N+6) log(5)) = e^((N+2) log(4)+(2 N+6) log(5)):
-1100000 107^N+1413867 e^(2 log(5) N+log(4) N)+3 e^(log(4) (N+2)+log(5) (2 N+6)) = 0

Divide both sides by 107^N:
2163867 107^(-N) e^((log(4)+2 log(5)) N)-1100000 = 0

Bring 2163867 107^(-N) e^(N (log(4)+2 log(5)))-1100000 together using the common denominator 107^N:
-107^(-N) (1100000 107^N-2163867 e^((log(4)+2 log(5)) N)) = 0

Multiply both sides by -1:
107^(-N) (1100000 107^N-2163867 e^((log(4)+2 log(5)) N)) = 0

Split into two equations:
107^(-N) = 0 or 1100000 107^N-2163867 e^((log(4)+2 log(5)) N) = 0

107^(-N) = 0 has no solution since for all z element R, 107^z>0:
1100000 107^N-2163867 e^((log(4)+2 log(5)) N) = 0

Divide both sides by e^(N (log(4)+2 log(5))):
1100000 107^N e^((-log(4)-2 log(5)) N)-2163867 = 0

107^N e^(N (-log(4)-2 log(5))) = e^(log(107^N)) e^(log(e^(N (-log(4)-2 log(5))))) = e^(N log(107)) e^((N (-log(4)-2 log(5))) log(e)) = e^(N log(107)+N (-log(4)-2 log(5))):
1100000 e^(log(107) N+(-log(4)-2 log(5)) N)-2163867 = 0

Add 2163867 to both sides:
1100000 e^(log(107) N+(-log(4)-2 log(5)) N) = 2163867

Divide both sides by 1100000:
e^(log(107) N+(-log(4)-2 log(5)) N) = 2163867/1100000

Take the natural logarithm of both sides:
log(107) N+(-log(4)-2 log(5)) N = log(2163867/1100000)

Expand and collect in terms of N:
(-log(4)-2 log(5)+log(107)) N = log(2163867/1100000)

Divide both sides by -log(4)-2 log(5)+log(107):
Answer: | N = (log(2163867/1100000))/(-log(4)-2 log(5)+log(107))=10

Sep 24, 2016
 #2
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Sep 24, 2016
 #1
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Sep 24, 2016

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