6.2.1 Laws of Logarithms

Laws of Logarithms

$a$	$b$	$\log_a$	$\log_b$	$\log(ab)$	$\log(a) + \log(b)$
2.1	6	0.3222	0.7782	1.1004	1.1004
3	3	0.4771	0	0.4771	0.4771
$\pi$	2.4	0.4971	0.3802	0.8774	0.8774
$\sqrt{2}$	$\sqrt{3}$	0.1505	0.4225	0.5730	0.5730

Key Result

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\log(ab) \equiv \log(a) + \log(b)

This suggests that:

Conversely,

10^a \times 10^b = 10^{a+b}

This just says the power of two things with the same base multiplied together is simply those two powers added.

Logarithms: Simplifying and Log Laws

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Examples:

Simplifying a Logarithm:

\log_{10}(10 \times 1000) = \log_{10}(10^1 \times 10^3) = \log_{10}(10^4) = 4

Simplifying into a Single Logarithm:

\log_{3}(10) + \log_{3}(5) = \log_{3}(10 \times 5) = \log_{3}(50)

\log_{5}(20) + \log_{5}\left(\frac{1}{4}\right) = \log_{5}\left(20 \times \frac{1}{4}\right) = \log_{5}(5) = 1

\log_{10}(20) + \log_{10}(20) = \log_{10}(20 \times 20) = \log_{10}(400)

Summary of Log Laws

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Addition Rule:

\log(a) + \log(b) = \log(ab)

Subtraction Rule:

\log(a) - \log(b) = \log\left(\frac{a}{b}\right)

Power Rule:

b \log(a) = \log(a^b)

Proof of Power Rule

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Starting with:

b \log(a) = \log(a) + \log(a) + \dots + \log(a) \quad \text{(b times)}

Using the addition rule, this simplifies to:

= \log(a \times a \times a \times \dots \times a) \quad \text{(b times)}

= \log(a^b)\ \square

Laws of Logarithms

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$2^x \times 2^3 = 2^{x+3}$ (When we multiply two things together with the same base, we add powers) $2^x \div 2^y = 2^{x-y}$ (When we divide, we subtract the powers)

Using Laws of Logarithms

Express in the form $\log_a n$ :

\log_a(4) + \log_a(7) = \log_a(4 \times 7) = \log_a(28) \quad \text{(Addition Rule)}

\log_a(10) - \log_a(5) = \log_a\left(\frac{10}{5}\right) = \log_a(2) \quad \text{(Subtraction Rule)}

2\log_a(6) = \log_a(6^2) = \log_a(36) \quad \text{(Power Rule)}

\log_a(9) - \log_a\left(\frac{1}{3}\right) = \log_a(9 \times 3) = \log_a(27) \quad \text{(Subtraction Rule)}

\frac{1}{2}\log_a(25) + 2\log_a(3) = \log_a(25^{\frac{1}{2}}) + \log_a(3^2) = \log_a(5) + \log_a(9) = \log_a(45) \quad \text{(Power Rule, Addition Rule)}

\log_a(48) - \frac{3}{2}\log_a(2) - \frac{1}{2}\log_a(9)

= \log_a(48) - \log_a(2^3) - \log_a(3)

= \log_a\left(\frac{48}{8 \times 3}\right) = \log_a(2) \quad \text{(Subtraction Rule)}

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Example: Write as a sum of separate logarithms involving no indices:

\log(3a^2b^3)

A common mistake:

\log(3a^2b^3) = \log(3ab) + 10\log(ab) \quad \text{(Incorrect)}

Correct:

\log(3) + 2\log(a) + 3\log(b)

Note: The power of 10 is only attached to $b^3$ ; it should not be applied to all terms.

Simplification of Logarithmic Expressions

The correct way to simplify:

\log(3ab^9) = \log(3) + \log(a) + 10\log(b) \quad \text{(Addition Rule, Power Rule)}

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Example: Simplify $1 + \log_3(5)$ as a single logarithm: $1$ is not in logarithmic form.

\log_3(3) = 1

Therefore,

1 + \log_3(5) = \log_3(3 \times 5) = \log_3(15)

Express the following in terms of $p$ and $q$ :

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Given that $\log_a x = p$ and $\log_a y = q$ : 6. (i) $\log_a (xy)$

\log_a (xy) = \log_a (x) + \log_a (y) = p + q

(ii) $\log_a \left(\frac{x^2 \cdot x^3}{y}\right)$

\log_a \left(\frac{x^2 \cdot x^3}{y}\right) = \log_a (x^2) + \log_a (x^3) - \log_a (y)

= 2\log_a (x) + 3\log_a (x) - \log_a (y)

= 2p + 3p - q = 5p - q

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Examples

Write as a single logarithms, simplifying where appropriate

Simplification Using Addition Rule:

\log(2) + \log(5) = \log(10) = 1

Combining Logs:

\ln(x) + 3 \ln(y) = \ln(x) + \ln(y^3) = \ln(x \cdot y^3)

(Here, Power Law and Addition Law are applied)

Simplifying Logarithmic Expressions

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Example Expression:

\frac{1}{2} \log(9) + 3\log(2)

Step 1: Apply the Power Rule

\frac{1}{2} \log(9) = \log(9^{\frac{1}{2}}) \quad \text{and} \quad 3\log(2) = \log(2^3)

Step 2: Substitute the values

= \log(3) + \log(8)

Step 3: Apply the Addition Rule

= \log(3 \times 8) = \log(24)

Logarithms: Correct Method for Solving Equations

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Example Problem:

Given the equation:

\log(x - 5) - \log(x) = 7

Wrong Method:

Attempting to solve by "unlogging" each term separately:

10^{\log(x-5)} - 10^{\log(x)} = 10^7

Reason:

1 + 2 = 3 \quad \text{but} \quad 10^1 + 10^2 \neq 10^3

Correct Method:

Gather all logarithms into one term before "unlogging":

\log\left(\frac{x-5}{x}\right) = 7

Applying the Subtraction Rule:

\frac{x-5}{x} = 10^7

Cross-multiply:

x - 5 = 10^7x

Isolating $x$ :

-5 = 10^7x - x

-5 = 9999999x \quad \Rightarrow \quad x = \frac{-5}{9999999}

Laws of Logarithms Simplified Revision Notes for A-Level AQA Maths Pure

6.2.1 Laws of Logarithms

Laws of Logarithms

Key Result

Logarithms: Simplifying and Log Laws

Examples:

Summary of Log Laws

Proof of Power Rule

Laws of Logarithms

Using Laws of Logarithms

Express in the form $\log_a n$ :

Example: Write as a sum of separate logarithms involving no indices:

Simplification of Logarithmic Expressions

Express the following in terms of $p$ and $q$ :

Examples

Simplifying Logarithmic Expressions

Example Expression:

Logarithms: Correct Method for Solving Equations

Example Problem:

500K+ Students Use These Powerful Tools to Master Laws of Logarithms For their A-Level Exams.

Other Revision Notes related to Laws of Logarithms you should explore

Laws of Logarithms Simplified Revision Notes for A-Level AQA Maths Pure

6.2.1 Laws of Logarithms

Laws of Logarithms

Key Result

Logarithms: Simplifying and Log Laws

Examples:

Summary of Log Laws

Proof of Power Rule

Laws of Logarithms

Using Laws of Logarithms

Express in the form log⁡an\log_a nloga​n:

Example: Write as a sum of separate logarithms involving no indices:

Simplification of Logarithmic Expressions

Express the following in terms of ppp and qqq:

Examples

Simplifying Logarithmic Expressions

Example Expression:

Logarithms: Correct Method for Solving Equations

Example Problem:

500K+ Students Use These Powerful Tools to Master Laws of Logarithms For their A-Level Exams.

Other Revision Notes related to Laws of Logarithms you should explore

Express in the form $\log_a n$ :

Express the following in terms of $p$ and $q$ :