Eleven Standard >> Properties of logarithm

Properties of logarithm

When the base of the logarithm is 'e' then it is called the natural logarithm.

When the base of the logarithm is 10, then it is called the common logarithm.

Basic properties of logarithm:

1) \(a^{\log_{a}{x}}=x\), if x>0
2) \(\log_{a}{a^{x}}=x\), for all \(x \in R\)

Rule of logarithm:

Rule 1. \(\log_{a}{1}=0\)
           \(\Rightarrow a^{0}=1\)

Rule 2. \(\log_{a}{a}=1\)
           \(\Rightarrow a^{1}=a\)

If a, b, M, N are positive real numbrs, (a≠1, b≠1), the following rules are applicable

Rule 3. \(\log_{a}{(MN)}=\log_{a}{M}+\log_{a}{N}\)

Proof: Let \(\log_{a}{M}=x\) and \(\log_{a}{N}=y\)
 \(\Rightarrow a^{x}=M\) and \(a^{y}=N\)
  \(MN=a^{x}.a^{y}=a^{x+y}\)
 \(\log_{a}{MN}=x+y\)=\(\log_{a}{M}+\log_{a}{N}\)

Rule 4. \(\log_{a}{\Big(\frac{M}{N}\Big)}=\log_{a}{M}-\log_{a}{N}\)

Proof: \(\log_{a}{M}=x\), \(\log_{a}{N}=y\)
\(\Rightarrow a^{x}=M\), \(a^{y}=N\)
\(\frac{M}{N}=\frac{a^{x}}{a^{y}}=a^{x-y}\)
\(\Rightarrow\) \(\log_{a}{\Big(\frac{M}{N}\Big)}=x-y=\log_{a}{M}-\log_{a}{N}\)

Rule 5. \(\log_{a}{M^{n}}=n\log_{a}{M}\)

Proof: Let \(\log_{a}{M^{n}}=x\)
\(\Rightarrow a^{x}=M^{n}\) and \(\log_{a}{M}=y\Rightarrow a^{y}=M\)
Now \(a^{x}=M^{n}=(a^{y})^{n}=a^ny\)
\(\Rightarrow x=ny\)
\(\Rightarrow \log_{a}{M^{n}}=n\log_{a}{M}\)