Properties of Logarithms

Logarithms are exponents in disguise. Because $b^m \cdot b^n = b^{m+n}$, it follows that the logarithm of a product is the sum of the logarithms. This is the Product Rule: $\log_b(xy) = \log_b x + \log_b y$. Similarly, the Quotient Rule mirrors the division of powers: $\log_b(\frac{x}{y}) = \log_b x - \log_b y$. These rules allow us to 'break apart' complex expressions into simpler, additive or subtractive pieces, which is essential for solving high-level algebraic equations.

The Power Rule is perhaps the most powerful tool in the logarithmic toolkit. It states that $\log_b(x^p) = p \cdot \log_b x$. Essentially, an exponent inside a logarithm can be moved to the front as a multiplier. This 'brings down' variables from the exponent position, making them much easier to isolate and solve. When combined with the product and quotient rules, you can condense multiple terms into a single, compact logarithm.

Most calculators only have buttons for $\log$ (base 10) and $\ln$ (base $e$). To evaluate a logarithm with an unusual base, like $\log_3 7$, we use the Change-of-Base Formula: $\log_b a = \frac{\log_c a}{\log_c b}$. Here, $c$ can be any base you choose. By choosing base 10 or $e$, you can solve any logarithmic value using a standard scientific calculator. This formula bridges the gap between abstract theory and numerical calculation.