# Solving Weird Exponential Equations

I am working on my math homework when I encountered a difficult problem. I simplified the equation and substituted smaller numbers to get this:

$n*2^n>10$

I have tried standard algebraic methods with logarithms, but I could not get them to work. Researching online, I came across the Lambert W function, but I know I don't need it to get the answer, as the math class I am taking is not that advanced. I strongly prefer not to use it.

If anyone can figure out the answer and explain, I would greatly appreciate it.

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If $n$ must be an integer, this equation holds for all $n\ge 3$. – vadim123 Jul 5 '14 at 23:58
I think the only way to solve that equation is to use the Lambert W function or you could just try integers (if $n$ is an integer) to see what works. As @vadim123 states $n\geq 3$ since the actual solution is approximately: $n>2.1906$. – Jay Jul 5 '14 at 23:58
@vadim123, how did you figure that out? – alexwho314 Jul 6 '14 at 0:23
@Jay, I would like to just try integers, but my teacher would probably prefer a better method than guess and check. – alexwho314 Jul 6 '14 at 0:25
@alexwho314 I don't think there is any other method that's much better than guessing and checking. You could use various numerical methods but that's hardly better. I'm fairly sure there isn't another way of solving an equation of the form $x a^x=b$ with elementary functions. Unless you've made an error in getting that equation then I can't see an obvious way of doing this without guessing and checking (and without using the Lambert W function). – Jay Jul 6 '14 at 0:31

Do you remember how to graph exponential and reciprocal functions by hand? If so, rearrange to:

$$2^n > \frac{10}{n}$$

The inequality is reversed if $n<0$, but it is obvious that no such solutions exist, so we ignore that case.

Then sketch the graphs by hand, note the intersection is around $n=2$, and test values $n=2$ and $n=3$ manually.

If you don't remember how to graph exponential and reciprocal functions, you can use numerical methods to approximate the point of intersection. Near zero, $2^n \approx \ln{2} \cdot (1+n+\frac{n^2}{2})$ (the first three terms of the Maclaurin sequence).

Using the well known approximation $\ln{2} \approx 0.7$, we can find the intersection by solving the polynomial:

$$$$\begin{split} 10 &\approx 0.7 \left(n+n^2+\frac{n^3}{2}\right) \newline 0 &\approx 7n^3 + 14n^2 + 14n - 200 \end{split}$$$$

Use a standard scientific calculator to solve the above, to obtain $n \approx 2.3$.

We then, once again, check $n=2$ and $n=3$ manually.

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Thank you very much. As much as I would like to avoid a graphical method of solving, that must be what I have to do. – alexwho314 Jul 6 '14 at 15:55