Logarithm Study Guide

📖 Core Concepts Logarithm – the exponent you need to raise a base \(b\) to get a number \(x\): \(\log{b}x = y \iff b^{y}=x\). Inverse relationship – exponentiation and logarithms undo each other; the graph of \(\log{b}x\) is the reflection of \(b^{x}\) across \(y=x\). Common bases – base 10 (common log, \(\log\) in engineering), base \(e\) (natural log, \(\ln\)), base 2 (binary log, \(\log{2}\) in CS). Logarithmic identities – product, quotient, power, root, and change‑of‑base formulas that let you rewrite logs in convenient forms. Derivative – \(\displaystyle \frac{d}{dx}\log{b}x = \frac{1}{x\,\ln b}\); for the natural log, \(\frac{d}{dx}\ln x = \frac{1}{x}\). Integral – \(\displaystyle \int \ln x\,dx = x\ln x - x + C\); also \(\displaystyle \ln t = \int{1}^{t}\frac{1}{x}\,dx\). --- 📌 Must Remember Product law: \(\log{b}(xy)=\log{b}x+\log{b}y\). Quotient law: \(\log{b}\!\left(\frac{x}{y}\right)=\log{b}x-\log{b}y\). Power law: \(\log{b}(x^{p}) = p\,\log{b}x\). Root law: \(\log{b}\!\left(\sqrt[p]{x}\right)=\frac{1}{p}\log{b}x\). Change‑of‑Base: \(\displaystyle \log{b}x = \frac{\log{k}x}{\log{k}b}\) (use \(k=10\) or \(e\) on calculators). Derivative: \(\displaystyle \frac{d}{dx}\log{b}x = \frac{1}{x\ln b}\). Number of decimal digits: \(\text{digits}(x)=\lfloor\log{10}x\rfloor+1\). Decibel power: \(L{\text{dB}} = 10\log{10}\!\left(\frac{P}{P{0}}\right)\). Decibel voltage: \(L{\text{dB}} = 20\log{10}\!\left(\frac{V}{V{0}}\right)\). pH definition: \(\text{pH} = -\log{10}[ \mathrm{H}^{+} ]\). Information (bits): \(I = -\log{2}p\); information (nats): \(I = -\ln p\). Algorithmic cost: Divide‑and‑conquer → \(\Theta(\log n)\); Merge sort → \(\Theta(N\log N)\); Binary search → \(\log{2}N\) comparisons. --- 🔄 Key Processes Changing the base of a log Choose a calculator‑friendly base \(k\) (10 or \(e\)). Compute \(\log{b}x = \frac{\log{k}x}{\log{k}b}\). Simplifying a log expression Break products → sum, quotients → difference, powers → coefficient, then combine using the above identities. Differentiating a log function Identify the base \(b\). Apply \(\frac{d}{dx}\log{b}x = \frac{1}{x\ln b}\). Computing algorithmic complexity Recognize a divide‑and‑conquer recurrence → \(T(n)=aT(n/b)+f(n)\) → solution often contains \(\log{b} n\). Applying log‑likelihood Take \(\ln\) of a product of independent likelihoods → sum of logs → easier maximisation. --- 🔍 Key Comparisons Common log vs. Natural log vs. Binary log Common (\(\log\) base 10): used in engineering, scientific notation, digit‑count. Natural (\(\ln\) base \(e\)): appears in calculus, continuous growth, entropy formulas. Binary (\(\log{2}\)): measures information in bits, appears in CS algorithm analysis. Product law vs. Power law Product: \(\log{b}(xy)=\log{b}x+\log{b}y\). Power: \(\log{b}(x^{p}) = p\,\log{b}x\) – a special case of the product law applied repeatedly. Derivative of \(\ln x\) vs. derivative of \(\log{b}x\) \(\displaystyle \frac{d}{dx}\ln x = \frac{1}{x}\). \(\displaystyle \frac{d}{dx}\log{b}x = \frac{1}{x\ln b}\) – just a constant scaling of the natural‑log derivative. --- ⚠️ Common Misunderstandings “\(\log x\) always means natural log.” – Context matters: math textbooks usually, engineering/science often means base 10, CS often means base 2. Confusing \(\log{b} (x^{p})\) with \((\log{b} x)^{p}\). The exponent stays outside the log: \(\log{b}(x^{p}) = p\log{b}x\). Assuming \(\log{b}(-x)\) is defined. Logarithms are only defined for positive arguments (real‑valued case). Treating the change‑of‑base formula as an equality for any \(k\), including \(k=1\). Base 1 is invalid; \(k\neq1\) and \(k>0\). --- 🧠 Mental Models / Intuition “Log = what you need to undo exponentiation.” Imagine climbing a ladder (exponentiation); the log tells you how many steps you climbed. Graph reflection: Visualize the exponential curve reflected over the line \(y=x\); that mirror image is the log curve. Digit count analogy: The integer part of \(\log{10}x\) tells you the number of zeros you’d write before the decimal point—hence digit count. Information bits: Each halving of probability adds 1 bit; each ten‑fold decrease adds \(\log{2}10\approx3.32\) bits. --- 🚩 Exceptions & Edge Cases Base \(0<b<1\): \(\log{b}x\) is decreasing (opposite monotonicity of \(b>1\)). Complex logarithm: Multi‑valued; principal value restricts argument to \(-\pi<\varphi\le\pi\). Discrete logarithm: Defined only in finite cyclic groups; no closed‑form formula like the real‑valued case. --- 📍 When to Use Which Base selection for calculations: Use \(\ln\) when calculus (derivatives, integrals) is involved. Use \(\log\) (base 10) for scientific‑scale problems (pH, decibels, digit counts). → If a problem mentions “bits” or “information,” pick \(\log{2}\). Change‑of‑Base: When a calculator only has \(\log\) and \(\ln\), apply the formula with \(k=10\) or \(e\). Algorithm analysis: Treat any recurrence that repeatedly halves a problem size as yielding a \(\log{2}\) term. --- 👀 Patterns to Recognize Log‑sum‑exp pattern: When you see \(\ln(e^{a}+e^{b})\), think “LSE” – it simplifies to \(\max(a,b)+\ln(1+e^{-|a-b|})\). Exponent‑log duality: Any product inside a log often signals you can turn it into a sum outside; likewise, a power inside a log signals a coefficient outside. Complex‑log branch cut: Problems involving negative real numbers often require specifying the principal value (cut along the negative real axis). --- 🗂️ Exam Traps Choosing the wrong base for a given context – e.g., using \(\log{10}\) in a CS “bits” question leads to a factor of \(\log{2}10\). Dropping the absolute value in \(\log|x|\) when differentiating – derivative of \(\ln|x|\) is \(1/x\), but forgetting the absolute value can cause sign errors for \(x<0\) (though real logs are undefined there, some conventions appear in antiderivatives). Misapplying the change‑of‑base formula with \(k=1\) – results in division by zero. Assuming \(\log{b}(x^{p}) = (\log{b}x)^{p}\) – leads to wildly incorrect numerical answers. Confusing decibel formulas – power ratios use factor 10, voltage ratios use factor 20; mixing them swaps the factor and yields a 6 dB error. ---