Q.State and prove the three fundamental laws of logarithms (the product, quotient and power laws), taking a common base $a$.

Each law is proved by writing the numbers as powers of the base and using the corresponding law of indices.

Let $M$ and $N$ be positive numbers and $a$ a valid base. Put $\log_{a}M = x$ and $\log_{a}N = y$, which by definition means

Product Law: $\log_{a}(MN) = \log_{a}M + \log_{a}N$.

Writing this in logarithmic form: $\log_{a}(MN) = x + y = \log_{a}M + \log_{a}N.$ ∎

Quotient Law: $\log_{a}(M/N) = \log_{a}M - \log_{a}N$.

In logarithmic form: $\log_{a}(M/N) = x - y = \log_{a}M - \log_{a}N.$ ∎

Power Law: $\log_{a}(M^{p}) = p\log_{a}M$.

In logarithmic form: $\log_{a}(M^{p}) = px = p\log_{a}M.$ ∎ …

Consistency check with fundamental values. Setting $N = 1$ in the product law gives $\log_{a}(M\times1) = \log_{a}M + \log_{a}1$, which forces $\log_{a}1 = 0$ — consistent with §2. Setting $M = N$ in the quotient law gives $\log_{a}(M/M) = \log_{a}1 = 0 = \ …

A common shortfall in the proof is asserting the laws without introducing $M = a^{x}$, $N = a^{y}$ — the proof is the reduction to the index laws, so omitting that step leaves the statement unproved. Another is misremembering the power law as $\log_ …