Solve for n.....

2.2244*1.01^n  = 2.2244 + 1.01^n

2.2244* 1.01^n  - 1.10^n  = 2.2244

1.01^n(2.2244 - 1)  = 2.2244

1.01^n ( 1.2244) = 2.2244

1.01^n = 2.2244 / 1.2244         take the log of both sides

log 1.01^n  = log [ 2.2244/1.2244]

n*log 1.01  = log [ 2.2244/1.2244]

n = log [ 2.2244/1.2244] / log [1.01]  = about  60.0016519101031466

2.2244*1.01^n=2.2244 + 1.01^n

Solve for n over the real numbers:
2.2244 1.01^n = 2.2244+1.01^n

2.2244 1.01^n = 5561 4^(-1-n) 25^(-2-n) 101^n and 2.2244+1.01^n = 5561/2500+(101/100)^n:
5561 4^(-1-n) 25^(-2-n) 101^n = 5561/2500+(101/100)^n

4^(-1-n) 25^(-2-n) 101^n = e^(log(4^(-1-n))) e^(log(25^(-2-n))) e^(log(101^n)) = e^((-1-n) log(4)) e^((-2-n) log(25)) e^(n log(101)) = exp((-1-n) log(4)+(-2-n) log(25)+n log(101)):
5561 exp(log(4) (-1-n)+log(25) (-2-n)+log(101) n) = 5561/2500+(101/100)^n

Subtract 5561/2500+(101/100)^n from both sides:
-5561/2500-(101/100)^n+5561 exp(log(4) (-1-n)+log(25) (-2-n)+log(101) n) = 0

Factor 4^(-n), 25^(-n) and constant terms from the left hand side:
4^(-1-n) 25^(-2-n) (3061 101^n-5561 100^n) = 0

Split into three equations:
4^(-1-n) = 0 or 25^(-2-n) = 0 or 3061 101^n-5561 100^n = 0

4^(-1-n) = 0 has no solution since for all z element R, 4^z>0:
25^(-2-n) = 0 or 3061 101^n-5561 100^n = 0

25^(-2-n) = 0 has no solution since for all z element R, 25^z>0:
3061 101^n-5561 100^n = 0

Divide both sides by 101^n:
3061-5561 (100/101)^n = 0

Subtract 3061 from both sides:
-5561 (100/101)^n = -3061

Divide both sides by -5561:
(100/101)^n = 3061/5561

Take the logarithm base 100/101 of both sides:
Answer: | n = (log(5561/3061))/(log(101/100))          or n=60

In equations like this, there is always a UNIQUE solution that will render both sides equal:

Example: 5 X 7.5^n=5 + 7.5^n, where n=.110746546628........and so on.

Can also be done as follows:

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