The Log Is the Exponential in a Mirror

You'll learnto see the logarithm as the exponential reflected across the line y = x.

Draw y = 2ˣ. Fold the plane along the diagonal y = x. The curve that lands on the other side is log₂x — same points, coordinates swapped. Every fact about the log falls out of that one fold: its domain, its wall at x = 0, its refusal to level off. Watch the fold happen, and watch the reciprocal impostor 1/2ˣ fail the audition.

9 STEPS · 6 QUICK CHECKS · y = 2ˣ ⇄ y = log₂x · v2

y = 2ˣ ⇄ log₂x

Before you start

What you're looking at

The curve y = 2^x and the diagonal y = x, with a point like (3, 8) about to become (8, 3).

The question

What happens to each point's coordinates when the plane folds along the diagonal?

Watch for

Coordinates swapping, the equal axis scaling that makes the fold land on the curve, and the reciprocal 1/2^x failing to match.

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