Topic Content:
- Laws of Logarithms
- Changing the Base of a Logarithm
1) \( \scriptsize \log_a MN = \log_a M \: +\: \log_aN \\ \rightarrow \scriptsize a^x \: \times \: a^y \\ \scriptsize = a^{x + y} \)
2) \( \scriptsize \log_a \normalsize\left (\frac{M}{N} \right ) \scriptsize = \log_a M \: – \: \log_aN \\ \rightarrow \scriptsize a^x \div a^y \\ \scriptsize = a^{x – y}\)
3) \(\scriptsize \log_{a} \left ( M^{n} \right ) = n \log_{a} M \\ \scriptsize \rightarrow \left ( a^{x} \right )^{y} = a^{xy} \)
(for any base “a” > 0, where \( \scriptsize a \neq 1\))
- Note that the three basic laws of logarithms are closely related to those of indices given earlier on.
- Note that \( \frac {\log M}{\log N} \scriptsize \neq \log M \; – \log N \)⚠️ (Very Important)
Special Logarithms
4) \(\scriptsize \log_{a}a = 1 \\ \scriptsize\rightarrow a^1 = a \)
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