Edexcel IAL - Pure Maths 2- 5.2 Laws of Logarithms- Study notes - New syllabus
Edexcel IAL – Pure Maths 2- 5.2 Laws of Logarithms -Study notes- New syllabus
Edexcel IAL – Pure Maths 2- 5.2 Laws of Logarithms -Study notes -Edexcel A level Physics – per latest Syllabus.
Key Concepts:
- 5.2 Laws of Logarithms
Laws of Logarithms
Logarithms are used to simplify calculations involving multiplication, division, and powers. The logarithm \( \log_a x \) is defined for:
\( a > 0,\ a \ne 1,\ x > 0 \)
The following laws apply only when these conditions are satisfied.
Fundamental Laws of Logarithms
| Law | Statement |
| Product law | \( \log_a(xy) \equiv \log_a x + \log_a y \) |
| Quotient law | \( \log_a\!\left(\dfrac{x}{y}\right) \equiv \log_a x – \log_a y \) |
| Power law | \( \log_a(x^k) \equiv k \log_a x \) |
| Reciprocal law | \( \log_a\!\left(\dfrac{1}{x}\right) \equiv -\log_a x \) |
| Base law | \( \log_a a = 1 \) |
Important Notes
- Logarithms turn multiplication into addition.
- Division becomes subtraction.
- Powers become multiplication.
- These laws allow simplification before calculation.
Example
Simplify:
\( \log_3 9 + \log_3 4 \)
▶️ Answer / Explanation
Use the product law:
\( \log_3 9 + \log_3 4 = \log_3(9 \times 4) \)
\( = \log_3 36 \)
Answer: \( \log_3 36 \)
Example
Simplify:
\( \log_5\!\left(\dfrac{25}{\sqrt{5}}\right) \)
▶️ Answer / Explanation
Rewrite using powers:
\( 25 = 5^2,\ \sqrt{5} = 5^{1/2} \)
\( \log_5\!\left(\dfrac{5^2}{5^{1/2}}\right) = \log_5(5^{3/2}) \)
Apply the power law:
\( = \dfrac{3}{2}\log_5 5 \)
Since \( \log_5 5 = 1 \):
Answer: \( \dfrac{3}{2} \)
Example
Simplify fully:
\( \log_a\!\left(\dfrac{x^3}{\sqrt{y}}\right) \)
▶️ Answer / Explanation
Rewrite the square root as a power:
\( \sqrt{y} = y^{1/2} \)
Apply the quotient law:
\( \log_a x^3 – \log_a y^{1/2} \)
Apply the power law:
\( 3\log_a x – \dfrac{1}{2}\log_a y \)
Final Answer:
\( 3\log_a x – \dfrac{1}{2}\log_a y \)
Example
Simplify:
\( \log_2(8x^2) – \log_2\!\left(\dfrac{1}{4x}\right) \)
▶️ Answer / Explanation
Use the quotient law:
\( \log_2\!\left(8x^2 \times 4x\right) \)
Simplify inside the logarithm:
\( \log_2(32x^3) \)
Write 32 as a power of 2:
\( 32 = 2^5 \)
Apply the product and power laws:
\( \log_2(2^5) + \log_2(x^3) = 5 + 3\log_2 x \)
Final Answer:
\( 5 + 3\log_2 x \)
Example
Simplify fully:
\( \log_a\!\left(\dfrac{\sqrt{x}}{a y^2}\right) \)
▶️ Answer / Explanation
Rewrite the square root as a power:
\( \sqrt{x} = x^{1/2} \)
Apply the quotient law:
\( \log_a x^{1/2} – \log_a(a y^2) \)
Apply the product law:
\( \log_a x^{1/2} – (\log_a a + \log_a y^2) \)
Apply power and base laws:
\( \dfrac{1}{2}\log_a x – (1 + 2\log_a y) \)
Simplify:
Final Answer:
\( \dfrac{1}{2}\log_a x – 2\log_a y – 1 \)