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Logarithmic Action (2)!

Interact with the applet below for a few minutes. . Prior to starting, feel free to move the brown and purple point wherever you'd like (on the x-axis.) You can move the green point anywhere you'd like as well. Note: In order to see the referenced black point that eventually appears, you may need to zoom out a bit.
In general, how can you express the sum of two logarithms as the logarithm of a single quantity?

Quick Demo (BGM: Andy Hunter)