Statement:
If
𝑎 ≠ 1, 𝑏 ≠ 1, and 𝑀 > 0, then:
\[ \log_a(M) = \frac {\log_b(M)}{\log_b(a)} \]
Proof:
Let,
\[ x = \log_a(M) \Rightarrow a^x = M \]
taking log base b on both sides:
\[ \log_b(M) = \log_b(a^x) \Rightarrow x \cdot \log_b(a) = \log_b(M) \]
that is,
\[x = \frac{\log_b(M)}{\log_b(a)}\]
therefore,
\[\log_a(M) = \frac{\log_b(M)}{\log_b(a)}\]
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