Logarithm laws | Year 10 | Ascent Library

01 / LEARN

Understand the idea.

Laws inherited from exponents

For positive $M, N$ and a fixed valid base $b$ , multiplication inside a logarithm becomes addition outside: ${log}_{b} (M N) = {log}_{b} M + {log}_{b} N$ . Division similarly becomes subtraction: ${log}_{b} (\frac{M}{N}) = {log}_{b} M - {log}_{b} N$ .

The power law is ${log}_{b} (M^{k}) = k {log}_{b} M$ for $M > 0$ . These laws come from exponent laws, so they apply to products, quotients and powers. There is no corresponding rule that splits the logarithm of a sum.

Check each original input

Before solving a logarithmic equation, require every original logarithm input to be positive. Combining logs may hide those individual conditions; solving the resulting polynomial can therefore produce candidates that must be rejected.

The change-of-base formula is ${log}_{b} M = \frac{{log}_{a} M}{{log}_{a} b}$ for $M > 0$ and positive bases $a, b \neq 1$ . It lets a calculator use a different base while keeping the logarithm value unchanged.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Combine logs

Simplify ${log}_{2} 12 - {log}_{2} 3$ .

Worked solution

1
Combine using the quotient law.
${log}_{2} 12 - {log}_{2} 3 = {log}_{2} (\frac{12}{3})$
The logarithms share a base and both original inputs are positive.
2
Simplify the input.
${log}_{2} (\frac{12}{3}) = {log}_{2} 4$
The quotient is $4$ .
3
Evaluate the exponent.
${log}_{2} 4 = 2$
Since $2^{2} = 4$ , the required exponent is $2$ .

Answer

EXAMPLE 02

Solve and reject an invalid value

Solve ${log}_{10} x + {log}_{10} (x - 9) = 1$ .

Worked solution

1
State the original domain.
$x > 0, x - 9 > 0 ⟹ x > 9$
Each separate logarithm needs a positive input.
2
Combine and undo the logarithm.
${log}_{10} [x (x - 9)] = 1 ⟹ x (x - 9) = 10$
The product law applies on the domain, and $10^{1} = 10$ .
3
Solve the polynomial equation.
$\begin{aligned} x^{2} - 9 x - 10 & = 0 \\ (x - 10) (x + 1) & = 0 \\ x & = 10 or - 1 \end{aligned}$
Factoring lists both algebraic candidates.
4
Reject the candidate outside the domain.
$x = 10, {log}_{10} 10 + {log}_{10} 1 = 1 + 0 = 1$
The value $- 1$ violates $x > 9$ ; $10$ satisfies the original equation.

Answer

$x = 10$

03 / YOUR TURN

Try it yourself.

Check your understanding

Simplify $2 {log}_{3} 5 - {log}_{3} 25$ .

Show answer & reasoning

The power law gives $2 {log}_{3} 5 = {log}_{3} (5^{2}) = {log}_{3} 25$ . Subtracting the same logarithm leaves $0$ .

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

Simplify ${log}_{2} 8 + {log}_{2} 4$ .
2 marks
Write ${log}_{5} 75 - {log}_{5} 3$ as one log and evaluate.
2 marks
Expand ${log}_{10} (\frac{x^{3}}{y})$ for $x, y > 0$ .
2 marks
Solve ${log}_{2} x + {log}_{2} 4 = 5$ .
2 marks
Solve ${log}_{10} (x - 1) + {log}_{10} (x + 1) = {log}_{10} 8$ .
3 marks
A student claims ${log}_{10} 2 + {log}_{10} 3 = {log}_{10} 5$ . Give the correct single log.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Simplify ${log}_{2} 8 + {log}_{2} 4$ .
2 marks
It equals ${log}_{2} 32 = 5$ .
5
Write ${log}_{5} 75 - {log}_{5} 3$ as one log and evaluate.
2 marks
${log}_{5} (\frac{75}{3}) = {log}_{5} 25 = 2$ .
2
Expand ${log}_{10} (\frac{x^{3}}{y})$ for $x, y > 0$ .
2 marks
Use quotient and power laws.
$3 {log}_{10} x - {log}_{10} y$
Solve ${log}_{2} x + {log}_{2} 4 = 5$ .
2 marks
${log}_{2} (4 x) = 5$ , so $4 x = 32$ and $x = 8$ .
$x = 8$
Solve ${log}_{10} (x - 1) + {log}_{10} (x + 1) = {log}_{10} 8$ .
3 marks
Domain $x > 1$ . The product gives $x^{2} - 1 = 8$ , so $x = \pm 3$ ; only $3$ is valid.
$x = 3$
A student claims ${log}_{10} 2 + {log}_{10} 3 = {log}_{10} 5$ . Give the correct single log.
3 marks
The product law multiplies $2$ and $3$ , rather than adding them.
${log}_{10} 6$

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Integer and fractional index laws; connecting fractional powers with roots; simplifying surds; operations with surds; rationalising denominators; exact and approximate values; exponential tables and graphs; growth/decay factors; comparing linear/quadratic/exponential patterns; exponential equations; logarithms as inverses; logarithm laws; interpreting logarithmic scales; graph transformations and parameter experiments; choosing and evaluating models.

Official curriculum source ↗

In this lesson

Understand the idea.

Laws inherited from exponents

Check each original input

See the method in action.

Combine logs

Combine using the quotient law.

Simplify the input.

Evaluate the exponent.

Solve and reject an invalid value

State the original domain.

Combine and undo the logarithm.

Solve the polynomial equation.

Reject the candidate outside the domain.

Try it yourself.

Check your understanding

Keep practising