cube root of N grows faster. When in doubt, just take a large number (like a million may be) and check. For example, if N=10^6, cube root will be 100, but log(N) would be ~19.93. Note that log is to the base 2.
keenan
@rasterize Plugging in a big number is an ok strategy, but doesn't always work! For instance, which function grows faster: $x^2 \sin(x)$, or $x^2 \cos(x)$? You can of course pick enormous numbers for which either one is bigger! To be absolutely sure your strategy of comparing values succeeds, you would need to determine, for instance, the largest value $x_{\mathrm{max}}$ for which the two functions are equal, and then check which one is bigger for a value $x$ larger than $x_{\mathrm{max}}$. Another strategy would be to take the ratio of the two functions and consider what happens as $x$ goes to $\infty$: does the ratio approach a value above 1 or below 1? Etc.
cube root of N grows faster. When in doubt, just take a large number (like a million may be) and check. For example, if N=10^6, cube root will be 100, but log(N) would be ~19.93. Note that log is to the base 2.
@rasterize Plugging in a big number is an ok strategy, but doesn't always work! For instance, which function grows faster: $x^2 \sin(x)$, or $x^2 \cos(x)$? You can of course pick enormous numbers for which either one is bigger! To be absolutely sure your strategy of comparing values succeeds, you would need to determine, for instance, the largest value $x_{\mathrm{max}}$ for which the two functions are equal, and then check which one is bigger for a value $x$ larger than $x_{\mathrm{max}}$. Another strategy would be to take the ratio of the two functions and consider what happens as $x$ goes to $\infty$: does the ratio approach a value above 1 or below 1? Etc.