Logarithms

What is a Logarithm?

A logarithm answers the question:

“To what exponent must a specific base be raised to get a certain number?”

If:

$$a^x = b$$

Then:

$$\log_a b = x$$

So, logarithm is just an exponent.

Example:

$$2^3 = 8 \Rightarrow \log_2 8 = 3$$

It tells us that 2 must be raised to 3 to get 8.

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When Are Logs Used?

• Whenever exponential growth or repeated multiplication is involved — like population growth, compound interest, computer science (algorithms), or time complexity.
• Logarithms "reverse" exponential growth.

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Logarithmic and Exponential Relationship

$$a^x = N \quad \Leftrightarrow \quad \log_a N = x$$

This equivalence is the core identity. Every log can be transformed into an exponential, and vice versa.

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Types of Logarithms

1. Common Logarithm

• Base = 10
• Written as $\log N$

2. Natural Logarithm

• Base = $e \approx 2.718$
• Written as $\ln N$

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Domain and Range

• Domain of $\log_b x$: $x > 0$ (logs of negative numbers or 0 are undefined in real numbers)
• Range: All real numbers $(-\infty, \infty)$

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Key Logarithmic Formulas

These are essential and frequently used in aptitude and competitive exams:

Basic Laws

1. Product Rule

$$\log_b (xy) = \log_b x + \log_b y$$

2. Quotient Rule

$$\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$$

3. Power Rule

$$\log_b (x^n) = n \cdot \log_b x$$

4. Root Rule

$$\log_b \left(\sqrt[n]{x}\right) = \frac{1}{n} \log_b x$$

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Conversion Formulas

5. Change of Base Formula

$$\log_b x = \frac{\log_k x}{\log_k b} \quad \text{(for any base } k \text{, typically 10 or e)}$$

6. Base-Exponent Swap

$$a^{\log_a x} = x \quad \text{and} \quad \log_a (a^x) = x$$

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Special Values to Remember

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Conceptual Tips and Common Mistakes

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Visual Understanding

• Graph of $y = \log_b x$:

  • Passes through (1, 0)
  • Increases slowly
  • Approaches $-\infty$ as $x \to 0^+$
  • Defined only for $x > 0$

• As base $b$ increases, the graph flattens.

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Real-World Applications

• pH Scale: $\text{pH} = -\log_{10}[H^+]$
• Decibels: $dB = 10 \log_{10} \left( \frac{P_1}{P_0} \right)$
• Complexity in CS: $O(\log n)$, $\log_2$ often used in binary search.

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Examples

Example 1:

Find $\log_2 16$

$$\log_2 16 = \log_2 (2^4) = 4$$

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Example 2:

Simplify:

$$\log_5 125 + \log_5 4$$ $$= \log_5 (125 \cdot 4) = \log_5 500$$

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Example 3:

If $\log_{10} 2 = 0.3010$, find $\log_{10} 8$

$$\log_{10} 8 = \log_{10} (2^3) = 3 \cdot \log_{10} 2 = 3 \cdot 0.3010 = 0.9030$$

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Example 4:

Find the value of $\log_3 81$

$$\log_3 (3^4) = 4$$

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Example 5:

Solve for $x$:

$$\log_2 x = 5 \Rightarrow x = 2^5 = 32$$