## PROPERTIES OF LOGARITHM

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:

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

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