Lesson 9, Topic 1
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# Laws of Logarithms

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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:

$$\scriptsize \log_{b} (a) = \normalsize \frac{\log_x (a)}{\log_x (b)}$$

error: