Worked example 1
Use a log law
Simplify log x + log 5.
- Use log a + log b = log(ab).
- Multiply the arguments.
Answer: log(5x).
Back to Higher Mathematics
Higher Mathematics
Log laws and equations
Simplify log expressions and solve exponential or logarithmic equations.
Before you start
- Know index laws.
- Use inverse operations confidently.
- Remember that logarithm arguments must be positive.
Explanation
Logarithms reverse exponential statements. Log laws mirror index laws and help combine, split or simplify expressions before solving equations.
When solving log equations, check any final values in the original equation because log arguments must be positive.
Method and rules
- log(ab) = log a + log b
- logab = log a − log b
- log(aⁿ) = n log a
- If log a = log b, then a = b for positive a and b.
Worked examples
Worked example 2
Solve a log equation
Solve log(x) + log(3) = log(12)
- Combine the left side: log(3x) = log(12)
- Equate arguments: 3x = 12
- Solve.
So: x = 4
Worked example 3
Solve an exponential
Solve 2ˣ = 16.
- Write 16 as a power of 2.
- 2ˣ = 2⁴
- Equate powers.
Answer: x = 4
Watch out
- Writing log a + log b as log(a + b).
- Ignoring positive-domain restrictions.
- Forgetting to check solutions after rearranging.
Exam reminder
Check log equation solutions in the original arguments. Any solution that makes a log input zero or negative is invalid.