Logarithmic Differentiation

Logarithmic differentiation is a powerful technique that simplifies the process of differentiating complex functions, especially those involving products, quotients, and exponents. Its effectiveness stems from the properties of logarithms, which can convert these complex operations into simpler sums and differences.

Let $M, N > 0$ and $b > 0, b \neq 1$. Let $p$ be any real number.

• Product Rule: $\log_b(MN) = \log_b(M) + \log_b(N)$
  This rule allows us to break down the logarithm of a product into a sum of logarithms.
• Quotient Rule: $\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)$
  This rule converts the logarithm of a quotient into a difference of logarithms.
• Power Rule: $\log_b(M^p) = p \log_b(M)$
  This rule brings an exponent down as a coefficient, which is extremely useful for functions of the form $f(x)^{g(x)}$.
• Change of Base Formula: $\log_b(M) = \frac{\log_c(M)}{\log_c(b)}$
  This allows conversion between different logarithm bases. For calculus, we often convert to the natural logarithm ($\ln x = \log_e x$) because its derivative is the simplest: $\frac{d}{dx}(\ln x) = \frac{1}{x}$.

A crucial extension for calculus is the chain rule applied to logarithms: If $u$ is a differentiable function of $x$, then $\frac{d}{dx}(\ln u) = \frac{1}{u} \cdot \frac{du}{dx} = \frac{u'(x)}{u(x)}$.