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Okay so I think I have a very trivial and short proof for the change of base rule but I'm worried it might be circular or wrong since I see it nowhere.

To prove $$\log_bx = \frac{\log_ax}{\log_ab}$$

Proving that $\log_ax = \log_ab*\log_bx$ will prove the above since it's the result of multiplying both sides by $\log_ab$ and hence the same relationship.

$\log_ab*\log_bx = \log_ab^{\log_bx}$ // Logarithm power rule.

$\log_ab^{\log_bx} = \log_ax$ // By definition $b^{\log_bx} = x$

Q.E.D. ?

Bernard
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Tim
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  • You could immediately say $\log_a b^{\log_b x} = \log_a x$ by definition, skipping the first step of your proof. – KM101 Jan 08 '19 at 20:28
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    Your proof is just fine. – Ethan Bolker Jan 08 '19 at 20:37
  • Nothing wrong with your proof. I tried to do a quick google search and most of the pages I found were of the very basic "just accept this rule cause I say so", but I was under the impression must explanations would be very similar to yours. – fleablood Jan 08 '19 at 20:49
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    You can start with $b^{\log_bx}=a^{\log_ax}$ and apply $\log_a$ to both sides but your proof is fine – Vasili Jan 08 '19 at 21:03

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