Year 12 Exponentials and logarithms: Logarithmic equations

Logarithmic equations often become manageable after you combine the logs into one expression and then convert back to exponential form.

Always check the domain from the original logarithms. Any log argument must stay positive.

Main method

State the domain restrictions from each log argument.
Use log laws to combine terms where possible.
Rewrite the equation in exponential form or solve the resulting algebra.
Reject any value that breaks the original domain.

Worked example

Problem: Solve

\log_{2} (x - 1) + \log_{2} (x - 3) = 3

Domain restriction: $x - 1 > 0$ and $x - 3 > 0$ , so $x > 3$ .
Use the product rule: $\log_{2} ((x - 1) (x - 3)) = 3$ .
Rewrite in exponential form: $(x - 1) (x - 3) = 2^{3} = 8$ .
Expand: $x^{2} - 4 x + 3 = 8$ .
Rearrange: $x^{2} - 4 x - 5 = 0$ .
Factor: $(x - 5) (x + 1) = 0$ .
Potential solutions are $x = 5$ or $x = - 1$ .
Check the domain $x > 3$ . Only $x = 5$ is valid.

Answer:

x = 5

Why extraneous answers appear

Logarithms carry domain restrictions from the start. When you combine logs and solve the resulting algebra, you are often solving an equation that has more solutions than the original logarithmic one. That is why checking the answer in the original domain is not optional. It is part of the method.

Worked example 2

Problem: Solve

\log_{10} (x + 4) - \log_{10} (x - 2) = 1

State the domain: $x + 4 > 0$ and $x - 2 > 0$ , so $x > 2$ .
Use the quotient rule: $\log_{10} ((x + 4) / (x - 2)) = 1$ .
Rewrite in exponential form: $(x + 4) / (x - 2) = 10$ .
Multiply both sides by $x - 2$ : $x + 4 = 10 x - 20$ .
Solve: $24 = 9 x$ , so $x = 8 / 3$ .
Check the domain. Since $8 / 3 > 2$ , the solution is valid.

Answer:

x = 8 / 3

Common mistakes

Skipping the domain restrictions at the start.
Using log laws across sums instead of products or quotients.
Converting to exponential form before simplifying the logs.
Accepting every algebraic root without checking the original equation.

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.

A quick final check for this topic is to ask whether the form of the answer matches the form of the question. Exact evaluations should usually stay exact. Solving questions should respect domain restrictions. Modelling questions should include sensible units, rounding, and interpretation. Those checks are short, but they are exactly the checks that stop a nearly correct solution from becoming an incorrect one.