Year 12 Exponentials and logarithms: Log laws

Log laws let you expand, condense and simplify logarithmic expressions without changing their value. They are most useful when every term has the same base.

The three laws to remember are product, quotient and power:

\log (M N)

\log (M / N)

and

\log (M^{k})

Main rules

$\log_{a} (M N) = \log_{a} (M) + \log_{a} (N)$
$\log_{a} (M / N) = \log_{a} (M) - \log_{a} (N)$
$\log_{a} (M^{k}) = k \log_{a} (M)$

Worked example

Problem: Simplify

2 \log_{5} (x) + \log_{5} (x - 4) - \log_{5} (25)

Use the power rule: $2 \log_{5} (x) = \log_{5} (x^{2})$ .
Now combine the first two terms with the product rule: $\log_{5} (x^{2}) + \log_{5} (x - 4) = \log_{5} (x^{2} (x - 4))$ .
Use the quotient rule to subtract $\log_{5} (25)$ : $\log_{5} (x^{2} (x - 4)) - \log_{5} (25) = \log_{5} (x^{2} (x - 4) / 25)$ .

Answer:

\log_{5} (x^{2} (x - 4) / 25)

, with domain

x > 4

When to expand and when to condense

Expanding is useful when you need to separate a product into simpler pieces or pull a power down in preparation for solving. Condensing is useful when several logarithms with the same base appear in one equation and you want to convert them into a single log. The direction matters. A good rule is to ask what the next line of working needs. If the next step is comparison or exponential conversion, one combined log is usually more helpful.

Worked example 2

Problem: Expand

\log_{3} (27 x^{2} / y)

Use the quotient rule first: $\log_{3} (27 x^{2} / y) = \log_{3} (27 x^{2}) - \log_{3} (y)$ .
Now split the product: $\log_{3} (27 x^{2}) = \log_{3} (27) + \log_{3} (x^{2})$ .
Use the power rule: $\log_{3} (x^{2}) = 2 \log_{3} (x)$ .
Since $27 = 3^{3}$ , we have $\log_{3} (27) = 3$ .
So the fully expanded form is $3 + 2 \log_{3} (x) - \log_{3} (y)$ .

Answer:

3 + 2 \log_{3} (x) - \log_{3} (y)

Common mistakes

Trying to turn $\log (a + b)$ into $\log a + \log b$ , which is invalid.
Ignoring the domain conditions created by every logarithm term.
Using laws across different bases.
Stopping after a partial simplification when one more rule would finish the job.

Revision focus

Exponential and logarithmic questions become much easier when you classify the task before starting. Ask whether you are evaluating a logarithm exactly, simplifying with log laws, solving an exponential equation, solving a logarithmic equation, or modelling growth and decay. Each of those tasks has a standard pattern, and confusion usually comes from mixing those patterns together rather than from the algebra itself.

Another high-value habit is to reverse-check the answer. If a logarithm equals a certain number, convert it back into exponential form and see whether the statement is true. If a model gives a time or population value, ask whether the rounding and units make sense in context. Domain restrictions, positive log arguments, and sensible growth factors are not side details in this topic. They are part of what makes the answer mathematically valid.

Practice links

Simplification of logarithmic expressions