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## Homework Statement

This isn't really a homework question, but I was considering a different model of library fines... where the fine at the end of the 1st late day is $0.01, and it increases geometrically with a common ratio of 2, so it goes like

$0.01, $0.03, $0.07, $0.15...

Then I was comparing this with a typical model of $0.10 additional fines with each day

$0.10, $0.20, $0.30, $0.40...

**2. Homework Equations /attempt at a solution**

Now, I can work out the day (let n denote this term) when the accumulated fines on the geometric model exceeds the arithmetic model by plotting the graph, where I get

0.1 + 0.1(n-1) < 0.01(2

^{n}-1)

10n - 2

^{n}+ 1 < 0

which has integer solutions for n = 6 or greater. My question is, how do you algebraically solve these sort of equations where the variable is involved twice, as an index and as a base? For example...

x

^{3}- e

^{x}= 0