The Exponent Is the Terrain

At first glance, seems larger because its base is . The calmer way to compare it is to put both expressions over the same base:

The factor changes the scale of the exponent, but not its polynomial degree. The expression still lives on a cubic exponent.

When the constants move

Now compare with using Big-O notation. Both mention the same cubic scale, but the sits in a different place:

Here . The first notation allows the exponent itself to be rescaled; the second only multiplies the finished exponential.

Take . It satisfies

but

so . See the limit-form definition of Big-O notation.

Here is the same separation as a picture. Literal exponentials become enormous quickly, so the vertical axis shows . Take as a representative of the outside- form and as a representative of the inside- form.

The two notations are therefore not interchangeable. First normalise the base and inspect the exponent; then check where lives. The exponent tells you which terrain you are walking through, while the position of tells you how wide the path may be.