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