1. Introduction
A logarithm tells us the power to which a base must be raised to get a number. If:
\( \log_b a = x \)
it means:
\( b^x = a \)
Logs help simplify big calculations, powers, growth problems, and scientific measurements.
2. Conditions for Logarithms
Before applying any law of logarithms, these conditions must be true:
- Base \(b > 0\) and \(b \neq 1\)
- Argument \(a > 0\)
- Logarithm is defined only for positive numbers
These conditions help avoid impossible or undefined values.
3. Product Rule
Law:
\( \log_b (xy) = \log_b x + \log_b y \)
This means the log of a product becomes a sum of logs.
3.1. Examples
1. \( \log_{10} (5 \cdot 20) = \log_{10} 5 + \log_{10} 20 \)
2. \( \log_2 (8 \cdot 4) = \log_2 8 + \log_2 4 = 3 + 2 = 5 \)
4. Quotient Rule
Law:
\( \log_b \left( \dfrac{x}{y} \right) = \log_b x - \log_b y \)
Log of a division becomes subtraction.
4.1. Examples
1. \( \log_{10} \left( \dfrac{100}{5} \right) = \log_{10} 100 - \log_{10} 5 \)
2. \( \log_2 \left( \dfrac{16}{2} \right) = \log_2 16 - \log_2 2 = 4 - 1 = 3 \)
5. Power Rule
Law:
\( \log_b (x^n) = n \log_b x \)
The exponent can be brought in front of the log.
5.1. Examples
1. \( \log_3 (3^4) = 4 \log_3 3 = 4 \)
2. \( \log_{10} (10^6) = 6 \log_{10} 10 = 6 \)
6. Combining Laws
Sometimes problems use all laws together.
6.1. Example
Simplify: \( \log_2 (16 \cdot 8^2) \)
\( = \log_2 16 + \log_2 (8^2) \)
\( = 4 + 2\log_2 8 = 4 + 2(3) = 10 \)
7. Common Mistakes
- Writing \( \log(x+y) = \log x + \log y \) (this is FALSE).
- Forgetting that logs work only on positive values.
- Applying laws before simplifying inside the log.
- Mixing base values incorrectly.
8. Quick Practice
Simplify using log laws:
- \( \log_5 (25 \cdot 5) \)
- \( \log_3 \left( \dfrac{81}{3} \right) \)
- \( \log_{10} (10^7) \)
- \( \log_2 (4^3 \cdot 8) \)
9. Summary
- Product Rule → sum of logs.
- Quotient Rule → difference of logs.
- Power Rule → exponent becomes a multiplier.
- Logs work only with positive arguments and valid bases.