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# PROPERTIES OF LOGARITHM

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Logarithms are very useful for lengthy calculation.

There are two types of logs: (1)  log   (2)  ln (Natural log)

(1) The base is 10 in log

(2) The base is e in natural log

Here, we will discuss the properties of log.

Some important properties with example are given below:

1. $$\displaystyle {{\log }_{a}}m+{{\log }_{a}}n={{\log }_{a}}\left( mn \right)$$
2. $$\displaystyle {{\log }_{a}}m-{{\log }_{a}}n={{\log }_{a}}\left( \frac{m}{n} \right)$$
3. $$\displaystyle {{\log }_{a}}{{m}^{n}}=n{{\log }_{a}}m$$
4. $$\displaystyle {{\log }_{{{a}^{b}}}}m=\frac{1}{b}{{\log }_{a}}m$$
5. $$\displaystyle {{\log }_{n}}m=\frac{{{\log }_{a}}m}{{{\log }_{a}}n}$$
6. $$\displaystyle {{\log }_{n}}m\times {{\log }_{m}}n=1$$
7. $$\displaystyle {{\log }_{p}}m\times {{\log }_{q}}p\times {{\log }_{n}}q={{\log }_{n}}m$$
8. $$\displaystyle \log 1=0$$
9. $$\displaystyle {{\log }_{m}}m=1$$
10. If $$\displaystyle {{\log }_{a}}x=y$$then $$\displaystyle x={{a}^{y}}$$

NOTE:

$$\displaystyle {{\log }_{a}}m+{{\log }_{a}}n\ne {{\log }_{a}}\left( m+n \right)$$

$$\displaystyle {{\log }_{a}}m-{{\log }_{a}}n\ne {{\log }_{a}}\left( m-n \right)$$

Example 1:

Simplify $$\displaystyle \log \left( \sqrt[3]{4}\,\times \sqrt[4]{3} \right)$$

Solution 1:

$$\displaystyle \log \left( \sqrt[3]{4}\times \sqrt[4]{3} \right)$$

$$\displaystyle \begin{array}{l}=\log \left( \sqrt[3]{4} \right)+\log \left( \sqrt[4]{3} \right)\\=\log {{\left( 4 \right)}^{{}^{1}\!\!\diagup\!\!{}_{3}\;}}+\log {{\left( 3 \right)}^{{}^{1}\!\!\diagup\!\!{}_{4}\;}}\\=\frac{1}{3}\log 4+\frac{1}{4}\log 3\\=\frac{1}{3}\log {{2}^{2}}+\frac{1}{4}\log 3\\=\frac{2}{3}\log 2+\frac{1}{4}\log 3\end{array}$$

Example 2:

Simplify $$\displaystyle 2{{\log }_{10}}5+{{\log }_{10}}8-{{\log }_{10}}2$$

Solution 2:

$$\displaystyle 2{{\log }_{10}}5+{{\log }_{10}}8-{{\log }_{10}}2$$

$$\displaystyle \begin{array}{l}={{\log }_{10}}{{5}^{2}}+{{\log }_{10}}8-{{\log }_{10}}2\\={{\log }_{10}}\left( \frac{25\times 8}{2} \right)\\={{\log }_{10}}\left( 100 \right)\\={{\log }_{10}}{{10}^{2}}\\=2{{\log }_{10}}10\\=2\times 1\\=2\end{array}$$