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) \(\scriptsize \log_{a}a = 1 \\ \scriptsize\rightarrow a^1 = a \)
5) \(\scriptsize \log_{a}1 = 0 \\ \scriptsize \rightarrow a^0 \)
6) \(\scriptsize \log_{a}\normalsize\left (\frac{1}{a} \right ) \scriptsize = -1 \\ \scriptsize \rightarrow a^{-1} = \normalsize \frac{1}{a} \)
7) \(\scriptsize \log_{a}\normalsize\left (\frac{1}{x} \right ) \scriptsize = -\log_a x \)
Changing the Base of a Logarithm:
Note that it is possible to change the base of a given logarithm to a more convenient base.
Suppose we wanted to find the value of the expression \(\scriptsize \log_2(50)\) Since 50 is not a rational power of 2, it is difficult to evaluate this without a calculator.
However, most calculators only directly calculate logarithms in base-10 and base e. So in order to find the value of \(\scriptsize \log_2(50)\) we must change the base of the logarithm first.
Suppose \(\scriptsize \log_{q} P = y, \; then \; q^y = P \)
Taking logs to base a of both sides of
\(\scriptsize q^y = P \)hence, \(\scriptsize \log_{a} q^y = \log_{a} P \)
i.e \(\scriptsize y \log_{a} q = \log_{a} P \)
i.e \(\scriptsize y = \normalsize \frac{\log_{a} P}{\log_{a} q} \)
\( \therefore \scriptsize \log_{q} P = \normalsize \frac{\log_{a} P}{\log_{a} q} \)
(where p, q are positive real numbers \( \neq \) 1).
If a = P then,
\( \scriptsize \log_{q} a = \normalsize \frac{\log_{a} a}{\log_{a} q} \) \( \scriptsize \log_{q} a = \normalsize \frac{1}{\log_{a} q} \)The change of base rule
The base of any logarithm can easily be changed by using the following rule:
\( \scriptsize \log_{b} (a) = \normalsize \frac{\log_x (a)}{\log_x (b)} \) See Example