Logarithm of Complex Numbers for JEE Main – Essential Guide

Introduction

Complex logarithms appear occasionally in JEE Main, often yielding surprising results like $i^i$ being real. Mastering these formulas provides quick solutions to otherwise tricky problems.

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The Core Formula (Memorize This!)

For $z = x + iy \neq 0$:

$\boxed{\log(z) = \ln|z| + i\arg(z)}$

Where:

• $|z| = \sqrt{x^2 + y^2}$ (modulus)
• $\arg(z)$ = angle in radians (principal value: $-\pi < \theta \leq \pi$)

Principal Value: The value when $\arg(z)$ is taken in $(-\pi, \pi]$.

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Essential Examples (JEE Favorites)

1. $\log(i)$

$i = e^{i\pi/2}$ ⇒ $\log(i) = 0 + i\cdot\frac{\pi}{2} = \frac{i\pi}{2}$

2. $\log(-1)$

$-1 = e^{i\pi}$ ⇒ $\log(-1) = 0 + i\pi = i\pi$

3. $\log(1+i)$

$1+i = \sqrt{2}e^{i\pi/4}$ ⇒ $\log(1+i) = \frac{1}{2}\ln 2 + \frac{i\pi}{4}$

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The $i^i$ Surprise – A Real Number!

$i^i = e^{i\log(i)} = e^{i\cdot\frac{i\pi}{2}} = e^{-\pi/2} \approx 0.2079$

Key Insight: $i^i$ is purely real despite having imaginary base and exponent!

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General Power Formula

For complex $z, w$: $\boxed{z^w = e^{w\log(z)}}$

Step-by-step:

1. Find $\log(z) = \ln|z| + i\arg(z)$
2. Multiply by $w$: $w\log(z)$
3. Compute $e^{\text{(result)}}$ using Euler's formula if needed

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JEE PYQs & Patterns Solved

PYQ 1: JEE Main 2019 Pattern

Find principal value of $i^i$

30-second solution: $\log(i) = i\pi/2$ $i^i = e^{i\cdot i\pi/2} = e^{-\pi/2}$

Answer: $e^{-\pi/2}$

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PYQ 2: JEE Advanced Pattern

Find $(-1)^i$

Solution: $\log(-1) = i\pi$ $(-1)^i = e^{i\cdot i\pi} = e^{-\pi}$

Answer: $e^{-\pi}$ (also real!)

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PYQ 3: Value Determination

If $\log(x+iy) = a+ib$, find $a, b$

Solution: $\log(x+iy) = \ln\sqrt{x^2+y^2} + i\tan^{-1}(y/x)$ So $a = \frac{1}{2}\ln(x^2+y^2)$, $b = \tan^{-1}(y/x)$ (with quadrant adjustment)

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Quick Reference Table

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Important Properties (with Caveats)

1. Product Rule (modulo $2\pi i$)

$\log(z_1z_2) = \log(z_1) + \log(z_2) + 2\pi i k$

2. Power Rule (for integer $n$)

$\log(z^n) = n\log(z) + 2\pi i k$

3. Conjugate Property

$\log(\bar{z}) = \overline{\log(z)} \ \text{if } \arg(z) \neq \pi$

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Special Results Worth Memorizing

1. $i^i = e^{-\pi/2}$ ≈ 0.2079 (real)
2. $(-1)^i = e^{-\pi}$ ≈ 0.0432 (real)
3. $2^i = \cos(\ln 2) + i\sin(\ln 2)$ (on unit circle)
4. $e^{i\pi} = -1$ (Euler's identity)

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Problem Types & Strategies

Type 1: Direct Evaluation

Example: Find $\log(1-i\sqrt{3})$ Strategy: Convert to polar form $re^{i\theta}$, then $\log(z) = \ln r + i\theta$

Solution: $1-i\sqrt{3} = 2e^{-i\pi/3}$ $\log(1-i\sqrt{3}) = \ln 2 - i\pi/3$

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Type 2: Complex Powers

Example: Find $(1+i)^{1-i}$ Strategy: Use $z^w = e^{w\log(z)}$

Solution: $\log(1+i) = \frac{1}{2}\ln 2 + i\pi/4$ $(1-i)\log(1+i) = \text{(expand)}$ $(1+i)^{1-i} = e^{\text{(result)}}$

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Type 3: Equation Solving

Example: Solve $e^z = 1+i$ Strategy: Take log: $z = \log(1+i) + 2\pi i k$

Solution: $z = \frac{1}{2}\ln 2 + i(\pi/4 + 2\pi k)$

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Common Mistakes to Avoid

1. Forgetting $2\pi i k$: In general solutions, include $k \in \mathbb{Z}$
2. Wrong quadrant: $\arg(x+iy) = \tan^{-1}(y/x)$ needs quadrant adjustment
3. Assuming real log rules: $\log(z_1z_2) \neq \log(z_1)+\log(z_2)$ exactly
4. Principal value range: Remember $(-\pi, \pi]$

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Practice Problems

Problem 1: Find $\log(-i)$

Solution: $-i = e^{-i\pi/2}$ $\log(-i) = 0 - i\pi/2 = -i\pi/2$

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Problem 2: Find $i^{2i}$

Solution: $i^{2i} = e^{2i\log(i)} = e^{2i\cdot i\pi/2} = e^{-\pi}$

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Problem 3: Solve $e^z = -2$

Solution: $z = \log(-2) = \ln 2 + i(\pi + 2\pi k)$

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Advanced Insight: Multi-valued Nature

The complete logarithm is: $\log(z) = \ln|z| + i(\arg(z) + 2\pi k), \ k \in \mathbb{Z}$

JEE typically uses principal value ($k=0$), but be aware of the general form.

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Exam Strategy

1. Recognize pattern: $\log(i), \log(-1), i^i$ appear frequently
2. Use polar form: $z = re^{i\theta}$ simplifies everything
3. For $z^w$: Always use $e^{w\log(z)}$
4. Check domain: $z \neq 0$ for $\log(z)$
5. Time: 1-2 minutes max for these problems

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Final Tip: Remember $i^i = e^{-\pi/2}$ as a surprising real number — a favorite JEE fact!

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Related: Complex Powers | Euler's Formula