Maths > Logarithms and Properties > 1.0 Introduction
Logarithms and Properties
1.0 Introduction
2.0 Properties of a logarithmic functions
3.0 Relation between common logarithm $\left( {{{\log }_{10}}x} \right)$ and Natural logarithm $\left( {{{\log }_e}x} \right)$
2.1 Proofs of all the above properties
2.2 System of logarithms
3.0 Sample Questions
4.0 Logarithmic Inequalities
1.2 Case 2: When $a=0$
2.1 Proofs of all the above properties
2.2 System of logarithms
Suppose the logarithm ${\log _a}N = x$ is defined.
Then, $$\begin{equation} \begin{aligned} {\log _a}N = x \\ {0^x} = N \\\end{aligned} \end{equation} $$
or $$0 = N$$
Example:
$$\begin{equation} \begin{aligned} {\log _0}2 = x \\ {0^x} = 2\;or\;0 = 2 \\\end{aligned} \end{equation} $$
Similarly,
$$\begin{equation} \begin{aligned} {\log _0}3 = x \\ {0^x} = 3\;or\;0 = 3 \\\end{aligned} \end{equation} $$
The above two example cannot be possible.
So for $a=0$, the logarithm ${\log _a}N = x$ is not defined.
