Proving 'Change of Base' Rule

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You're my hero, I've been looking for this for 30min, just what I needed thanks.

paulhaso
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Somebody Correct me if I am wrong. When he replaced x for "log a^b" and simultaneously took "log c^a" to the bottom of the other side, should "log c^a" not have changed into "log a^c" ??? If not, why?

shantahalder