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$ \log_\beta \alpha \log_\gamma \beta = \log_\gamma \alpha$

proof is given here -in 链式 section

proof from wiki

I cannot find this formula's english name(it is not about derivative as its name suggests)nor any version with further explaination except this. I don't understand the first line of proof, please explain. Thanks.

Blue
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    Would you prefer an answer in English or Chinese? – J.G. May 04 '22 at 11:35
  • Both are ok to me. – Blue May 04 '22 at 11:37
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    You need to understand two facts, (i) $\log_\gamma\beta=\frac{\ln\beta}{\ln\gamma}$ and (ii) when the proof rewrites the original logarithms using (i), it confusingly swaps the order. Do you know (i), and if so how to prove it? Proof of (i):$$e^{\ln\beta}=\beta=\gamma^{\log_\gamma\beta}=e^{\ln\gamma\times\log_\gamma\beta}\implies\ln\beta=\ln\gamma\times\log_\gamma\beta.$$ – J.G. May 04 '22 at 11:40
  • It's usually referred to in English as the Change of Base formula, and is usually written as $$log_ab=\frac{log_cb}{log_ca}$$ in school exam formula books. – David Quinn May 04 '22 at 16:45

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