Properties of logarithms: product, quotient and power rules

Warm-up

Three quick questions from earlier lessons.

Discover

Use the fact that $\log_2 x$ is the exponent of 2. Compare $\log_2 4 + \log_2 8$ with $\log_2 32$ .

Why are they equal? Hint: $4 \cdot 8 = 32$ , and $2^2 \cdot 2^3 = 2^{2+3}$ .

Key idea

Logarithms turn multiplication into addition, because exponents add when powers multiply.

\log_b(MN) = \log_b M + \log_b N

;

\log_b \dfrac{M}{N} = \log_b M - \log_b N

;

\log_b(M^p) = p \log_b M

These hold for $M > 0$ and $N > 0$ .

Worked example

Evaluate $\log_6 4 + \log_6 9$ without a calculator.

= \log_6(4 \cdot 9)

Product rule.

Why is the first line true?

The log of a product is the sum of the logs
You can add inside the logarithm
The logs multiply

= \log_6 36

4 · 9 = 36.

= 2

6² = 36.

Your turn, guided

Your turn: evaluate $2\log_3 6 - \log_3 4$ . Fill in the missing line.

= \log_3 36 - \log_3 4

Power rule: 2 log₃ 6 = log₃ 36.

= \log_3 9

Quotient rule: 36 ÷ 4 = 9.

= 2

3² = 9.

Practice ★

Which expression is equal to $\log_{ 8 }(2 \cdot 7)$ ?
1. $\log_{ 8 } 2 + \log_{ 8 } 7$
2. $\log_{ 8 } 2 \cdot \log_{ 8 } 7$
3. $\log_{ 8 }(2 + 7)$
4. $7 \log_{ 8 } 2$
Which expression is equal to $\log_{ 7 }(4 \cdot 9)$ ?
1. $\log_{ 7 } 4 + \log_{ 7 } 9$
2. $\log_{ 7 } 4 \cdot \log_{ 7 } 9$
3. $\log_{ 7 }(4 + 9)$
4. $9 \log_{ 7 } 4$
Which expression is equal to $\log_{ 7 }(5 \cdot 3)$ ?
1. $\log_{ 7 } 5 + \log_{ 7 } 3$
2. $\log_{ 7 } 5 \cdot \log_{ 7 } 3$
3. $\log_{ 7 }(5 + 3)$
4. $3 \log_{ 7 } 5$

Evaluate $\log_{ 10 } 20 + \log_{ 10 } 5$ without a calculator.
Evaluate $\log_{ 10 } 40 + \log_{ 10 } 25$ without a calculator.
Evaluate $\log_{ 10 } 50 + \log_{ 10 } 20$ without a calculator.
Evaluate $\log_{ 10 } 25 + \log_{ 10 } 4$ without a calculator.
Evaluate $\log_{ 6 } 12 + \log_{ 6 } 18$ without a calculator.

Watch out!

Practice ★★

Evaluate $2\log_{ 2 } 6 - \log_{ 2 } 9$ without a calculator.
Evaluate $2\log_{ 3 } 6 - \log_{ 3 } 4$ without a calculator.
Evaluate $2\log_{ 5 } 15 - \log_{ 5 } 9$ without a calculator.
Evaluate $2\log_{ 3 } 18 - \log_{ 3 } 12$ without a calculator.
Evaluate $2\log_{ 3 } 15 - \log_{ 3 } 75$ without a calculator.

Write $\log_{ 3 } x + 2\log_{ 3 } 4$ as a single logarithm.
Write $\log_{ 6 } x + 2\log_{ 6 } 5$ as a single logarithm.
Write $\log_{ 7 } x + 2\log_{ 7 } 4$ as a single logarithm.

Expand $\log_{ 8 }\left(\dfrac{x^{ 4 }}{y^{ 5 }}\right)$ : write it with logarithms of $x$ and $y$ only.
Expand $\log_{ 5 }\left(\dfrac{x^{ 5 }}{y^{ 3 }}\right)$ : write it with logarithms of $x$ and $y$ only.
Expand $\log_{ 8 }\left(\dfrac{x^{ 2 }}{y^{ 5 }}\right)$ : write it with logarithms of $x$ and $y$ only.

Exit ticket

Three questions to check you have got it.

Evaluate $2\log_{ 2 } 6 - \log_{ 2 } 9$ without a calculator.
Evaluate $2\log_{ 3 } 6 - \log_{ 3 } 4$ without a calculator.
Evaluate $2\log_{ 5 } 15 - \log_{ 5 } 9$ without a calculator.

Summary card

Properties of logarithms ( $M, N > 0$ )

Product: $\log_b(MN) = \log_b M + \log_b N$ .
Quotient: $\log_b \frac{M}{N} = \log_b M - \log_b N$ .
Power: $\log_b(M^p) = p \log_b M$ .

Watch out: there is no rule for $\log_b(M + N)$ .

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