Logarithms

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Logarithms — Quick Facts

Key Definitions:

• Logarithm: If $a^x = b$, then $\log_a b = x$ (read as “log base $a$ of $b$ equals $x$”)
• Base: The number that is raised to a power
• Antilogarithm: The number corresponding to a given logarithm
• Characteristic: The integer part of a logarithm
• Mantissa: The fractional part of a logarithm

Important Logarithmic Identities:

Identity	Formula
Product Rule	$\log_a(mn) = \log_a m + \log_a n$
Quotient Rule	$\log_a(m/n) = \log_a m - \log_a n$
Power Rule	$\log_a(m^n) = n \cdot \log_a m$
Change of Base	$\log_a m = \frac{\log_b m}{\log_b a}$
Base Switch	$\log_a b = \frac{1}{\log_b a}$

⚡ Exam Tips for NDA:

• Common bases: 10 (common log), $e$ (natural log, written as $\ln$)
• $\log_a a = 1$ and $\log_a 1 = 0$ for any valid base $a$
• $\log 100 = 2$ because $10^2 = 100$
• When solving equations, convert to exponential form first

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding.

Logarithms — Study Guide

Understanding Logarithms:

A logarithm answers the question: “To what power must we raise a base to get a certain number?”

$$2^5 = 32 \iff \log_2 32 = 5$$

The base $2$ raised to the power $5$ gives $32$, so the logarithm of $32$ with base $2$ is $5$.

Types of Logarithms:

Type	Base	Written As	Common Use
Common Logarithm	10	$\log x$ or $\log_{10} x$	General computation
Natural Logarithm	$e \approx 2.71828$	$\ln x$ or $\log_e x$	Calculus, science
Binary Logarithm	2	$\log_2 x$	Computer science

Laws of Logarithms — Proofs:

1. Product Rule: Let $\log_a m = x \Rightarrow a^x = m$ and $\log_a n = y \Rightarrow a^y = n$ $$mn = a^x \cdot a^y = a^{x+y}$$ Taking log base $a$: $$\log_a(mn) = x + y = \log_a m + \log_a n \quad \blacksquare$$

2. Quotient Rule: $$\frac{m}{n} = \frac{a^x}{a^y} = a^{x-y}$$ $$\log_a(m/n) = x - y = \log_a m - \log_a n \quad \blacksquare$$

3. Power Rule: $$m^n = (a^x)^n = a^{nx}$$ $$\log_a(m^n) = nx = n \cdot \log_a m \quad \blacksquare$$

Change of Base Formula:

$$\log_a m = \frac{\log_b m}{\log_b a}$$

Derivation: Let $\log_a m = x \Rightarrow a^x = m$ Taking log base $b$ on both sides: $$\log_b(a^x) = \log_b m$$ $$x \cdot \log_b a = \log_b m$$ $$x = \frac{\log_b m}{\log_b a}$$ $$\therefore \log_a m = \frac{\log_b m}{\log_b a} \quad \blacksquare$$

Logarithmic Equations:

Type 1: $\log_a f(x) = \log_a g(x)$ $$\Rightarrow f(x) = g(x) \quad \text{(provided } f(x), g(x) > 0\text{)}$$

Type 2: $\log_a f(x) = b$ $$\Rightarrow f(x) = a^b$$

⚡ Important Domain Condition: For $\log_a x$ to be defined:

• $a > 0$ and $a \neq 1$ (base conditions)
• $x > 0$ (argument must be positive)

Common Student Mistakes:

• Confusing $\log(m+n)$ with $\log m + \log n$ — they are NOT equal
• Forgetting that $\log_a(bc) \neq \log_a b \cdot \log_a c$ — that’s the wrong formula
• Solving $\log x = 2$ as $x = 2$ instead of $x = 10^2 = 100$

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🔴 Extended — Deep Study (3mo+)

Comprehensive theory for thorough preparation.

Logarithms — Comprehensive Notes

Graph of Logarithmic Function:

The function $y = \log_a x$ has these properties:

Base Condition	Graph Shape	Key Behaviour
$a > 1$	Increasing curve	$y$ increases as $x$ increases
$0 < a < 1$	Decreasing curve	$y$ decreases as $x$ increases

Key Points on the Graph:

• Passes through $(1, 0)$ always — because $\log_a 1 = 0$
• Domain: $x > 0$
• Range: All real numbers $(-\infty, +\infty)$
• Y-axis ($x = 0$) is a vertical asymptote

Relationship Between Exponential and Logarithmic Functions:

$$y = a^x \iff x = \log_a y$$

They are inverse functions of each other. The graph of $y = \log_a x$ is the reflection of $y = a^x$ across the line $y = x$.

Solving Complex Logarithmic Equations:

Example 1: Solve: $\log_3(x+1) + \log_3(x-1) = 1$

Solution: Using product rule: $\log_3[(x+1)(x-1)] = 1$ $$\log_3(x^2 - 1) = 1$$ $$x^2 - 1 = 3^1 = 3$$ $$x^2 = 4$$ $$x = \pm 2$$

Checking domain: $x + 1 > 0 \Rightarrow x > -1$ and $x - 1 > 0 \Rightarrow x > 1$ $$\therefore x = 2 \text{ (only valid solution)}$$

Example 2: Solve: $\log_2(x+2) - \log_2(x-1) = 3$

Solution: Using quotient rule: $\log_2\left(\frac{x+2}{x-1}\right) = 3$ $$\frac{x+2}{x-1} = 2^3 = 8$$ $$x + 2 = 8x - 8$$ $$7x = 10$$ $$x = \frac{10}{7}$$

Checking domain: $x > 1$ (from $x - 1 > 0$) $$\frac{10}{7} \approx 1.43 > 1 \quad \checkmark$$

Logarithmic Inequalities:

Rule: When solving $\log_a f(x) > \log_a g(x)$:

• If $a > 1$: $f(x) > g(x)$ (inequality sign preserved)
• If $0 < a < 1$: $f(x) < g(x)$ (inequality sign reverses)

Example: Solve: $\log_2(x+3) > \log_2(2x-1)$

Since base $2 > 1$: $$x + 3 > 2x - 1$$ $$x < 4$$

With domain conditions: $x + 3 > 0 \Rightarrow x > -3$ and $2x - 1 > 0 \Rightarrow x > 0.5$ $$\therefore 0.5 < x < 4$$

Characteristic and Mantissa:

For $\log_{10} N$ where $N > 0$:

• Characteristic ($c$): Integer part, depends on number of digits before decimal
• Mantissa ($m$): Fractional part, found from log tables

Number Range	Characteristic	Example
$0.0001 \leq N < 0.001$	$-4$	$\log 0.0052 \approx -3 + \text{mantissa}$
$0.001 \leq N < 0.01$	$-3$
$0.01 \leq N < 0.1$	$-2$
$0.1 \leq N < 1$	$-1$	$\log 0.5 \approx -0.3010$
$1 \leq N < 10$	$0$	$\log 5 = 0.6990$
$10 \leq N < 100$	$1$	$\log 25 = 1.3979$
$100 \leq N < 1000$	$2$	$\log 365 = 2.5623$

Antilogarithm: If $\log N = x$, then $N = \text{antilog}(x)$ $$\text{antilog}(x) = 10^x$$

⚡ NDA High-Yield Patterns:

• Questions frequently test direct application of log laws
• Solving logarithmic equations appears in both Paper I and II
• Change of base formula is commonly used in computer science-related NDA questions
• Expect 2-3 questions directly from this topic
• Always check domain restrictions in logarithmic equations — it’s a favourite trap

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