Log Formula Without Calculator

What is the Log Formula?

The logarithm (log) is the inverse function of exponentiation. It answers the question: "To what power must a base number be raised to obtain a given number?" The general log formula is:

There are three common types of logarithms:

• Common logarithm (base 10): Used in many mathematical applications. Denoted as \( \log_{10} \) or simply \( \log \).
• Natural logarithm (base e): Used in calculus and higher mathematics. Denoted as \( \ln \).
• Binary logarithm (base 2): Used in computer science. Denoted as \( \log_2 \).

For example, \( \log_{10} 100 = 2 \) because \( 10^2 = 100 \).

How to Calculate Logarithms Without a Calculator

Calculating logarithms manually requires understanding the relationship between exponents and logarithms. Here's a step-by-step method to compute logarithms without a calculator:

Step 1: Understand the Logarithm Definition

The logarithm \( \log_b a \) asks, "To what power must \( b \) be raised to get \( a \)?" For example, \( \log_2 8 = 3 \) because \( 2^3 = 8 \).

Step 2: Use Known Logarithm Values

Memorize common logarithm values for bases 10 and natural logarithms (base e). For example:

• \( \log_{10} 1 = 0 \)
• \( \log_{10} 10 = 1 \)
• \( \log_{10} 100 = 2 \)
• \( \ln e \approx 1 \)
• \( \ln e^2 \approx 2 \)

Step 3: Apply Logarithm Properties

Use logarithm properties to simplify calculations:

• Product rule: \( \log_b (xy) = \log_b x + \log_b y \)
• Quotient rule: \( \log_b \left( \frac{x}{y} \right) = \log_b x - \log_b y \)
• Power rule: \( \log_b (x^y) = y \log_b x \)
• Change of base formula: \( \log_b a = \frac{\log_k a}{\log_k b} \) (where \( k \) is any positive number)

Step 4: Use the Change of Base Formula

If you only know logarithms for base 10 or natural logarithms, use the change of base formula to compute logarithms for other bases:

For example, to compute \( \log_2 8 \) using base 10 logarithms:

1. \( \log_2 8 = \frac{\log_{10} 8}{\log_{10} 2} \)
2. \( \log_{10} 8 \approx 0.9031 \)
3. \( \log_{10} 2 \approx 0.3010 \)
4. \( \log_2 8 \approx \frac{0.9031}{0.3010} \approx 3 \)

Step 5: Use Logarithmic Tables or Series Expansion

For more precise calculations, use logarithmic tables or series expansions like the Taylor series for \( \ln(1+x) \).

Common Log Examples

Here are some common logarithm examples and their solutions:

Example 1: Common Logarithm

Calculate \( \log_{10} 1000 \).

Solution: \( \log_{10} 1000 = 3 \) because \( 10^3 = 1000 \).

Example 2: Natural Logarithm

Calculate \( \ln e^5 \).

Solution: \( \ln e^5 = 5 \) because \( e^5 = e^5 \).

Example 3: Binary Logarithm

Calculate \( \log_2 16 \).

Solution: \( \log_2 16 = 4 \) because \( 2^4 = 16 \).

Example 4: Logarithm with Change of Base

Calculate \( \log_3 9 \) using the change of base formula.

Solution:

1. \( \log_3 9 = \frac{\log_{10} 9}{\log_{10} 3} \)
2. \( \log_{10} 9 \approx 0.9542 \)
3. \( \log_{10} 3 \approx 0.4771 \)
4. \( \log_3 9 \approx \frac{0.9542}{0.4771} \approx 2 \)

Logarithm Properties

Logarithms have several important properties that simplify calculations and solve equations:

Product Rule

Example: \( \log_{10} (2 \times 5) = \log_{10} 2 + \log_{10} 5 \approx 0.3010 + 0.6990 = 1 \)

Quotient Rule

Example: \( \log_{10} \left( \frac{100}{10} \right) = \log_{10} 100 - \log_{10} 10 = 2 - 1 = 1 \)

Power Rule

Example: \( \log_{10} (10^3) = 3 \log_{10} 10 = 3 \times 1 = 3 \)

Change of Base Formula

Example: \( \log_2 8 = \frac{\log_{10} 8}{\log_{10} 2} \approx \frac{0.9031}{0.3010} \approx 3 \)