1.3 Case 3: When $N<0$

1.1 Case 1: When $a=1$
1.2 Case 2: When $a=0$
1.3 Case 3: When $N<0$

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

Suppose the logarithm ${\log _a}N = x$ is defined.

Then, $$\begin{equation} \begin{aligned} {\log _a}N = x \\ {a^x} = N \\ {\left( { + ve\;number} \right)^x} = \left( { - ve\;number} \right) \\\end{aligned} \end{equation} $$

The above situation is never possible.

So for $N<0$, the logarithm ${\log _a}N = x$ is not defined.