1. Restriction on the Base: Why $a$ Cannot Be Negative

The base of a logarithm ($a$) must be positive ($a > 0$). Allowing a negative base ($a < 0$) leads to contradictions or complex numbers, making the logarithm function inconsistent for real numbers.

Contradiction Example (Using $a = -1$):

1. If we take an even exponent:
   $$(-1)^2 = 1 \implies 2 = \log_{-1} 1$$
2. If we take an even exponent where the result is the same:
   $$(-1)^4 = 1 \implies 4 = \log_{-1} 1$$

This suggests that $2 = 4$, which is not possible.

Inconsistency Example (Using different exponents):

1. $$2^2 = 4 \implies 2 = \log_2 4$$
2. $$(-2)^2 = 4 \implies 2 = \log_{-2} 4$$

If we try to use $(-2)^4 = 16$, then $4 = \log_{-2} 16$. The values of $\log_{-2} N$ would oscillate between real and complex numbers as $N$ changes.

Conclusion: To maintain a consistent and well-defined function in the set of real numbers, the base must be restricted to $a > 0$ and $a \neq 1$.

2. Converting Between Forms

The fundamental relationship is:

A. Exponential to Logarithmic Form

B. Logarithmic to Exponential Form