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Is there any way to isolate $n$ algebraically in this inequality? I've managed to find it graphically, but I'm lost on how I would apply algebra to this question.

$$10000+800n>10000\times1.05^n$$

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  • $\begingroup$ Apply logarithms to both sides, then subtract getting one side with the logs being > 0 is one way to go about it. $\endgroup$ – OLE Nov 29 '16 at 5:25
  • $\begingroup$ @ELO, this will not work, as you have a sum inside a log. There is no way to solve for $n$ analytically without heavy machinery like the Lambert function. The approximate positive solution is $18.8376245\ldots$. $\endgroup$ – vadim123 Nov 29 '16 at 5:33
  • $\begingroup$ If you're interested in integer domain, then a cubic approximation to RHS will suffice, and is analytically solvable. It looks like when simple interest at 8% compares to compound interest at 5%, so perhaps all you need is integers. $\endgroup$ – Macavity Nov 29 '16 at 5:47

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