Which of the numbers $99^{100}$ and $100^{99}$ is the larger one? Which of the numbers $99^{100}$ &amp; $100^{99}$ is the larger? Solve without using logarithms.
Note that 
$$\begin{align}
99^{100} &gt; 100^{99} &amp;\iff 99 \cdot 99^{99} &gt; 100^{99}
\\
&amp;\iff 99 &gt; (100/99)^{99}
\\
&amp;\iff 99 &gt; \left( 1 + \frac{1}{99}\right)^{99}
\end{align}$$

Since $(1 + \frac{1}{n})^n &lt; 3$ for all integers $n$, the above inequalities are all true.  Thus, $99^{100} &gt; 100^{99}$.  In general, you should expect that $x^y &gt; y^x$, whenever $y &gt; x$.
From experimenting with small numbers:

scala&gt; (0 to 5).map (x=&gt; (math.pow (x, x+1) - math.pow (x+1, x))).mkString ("; ")
res18: String = -1.0; -1.0; -1.0; 17.0; 399.0; 7849.0

scala&gt; (0 to 5).map (x=&gt; (math.pow (x, x+1), math.pow (x+1, x))).mkString ("; ")
res19: String = (0.0,1.0); (1.0,2.0); (8.0,9.0); (81.0,64.0); (1024.0,625.0); (15625.0,7776.0)


you can conclude, that the first one is growing faster than the second. Of course this is only an indication.