Combining logarithms means using three laws so that several log terms become one: add logs to multiply, subtract logs to divide, and move a coefficient up to become a power. All the logs must share the same base.
This lesson follows converting between exponential and logarithmic statements and prepares the ground for equations in exponential and logarithmic reasoning.
What are the three laws?
For a positive base a and positive numbers x and y:
- loga x + loga y = loga (xy)
- loga x − loga y = loga (x/y)
- n loga x = loga (xⁿ)
Each comes from index laws. Law 1 is the index law aᵖ × aᵠ = aᵖ⁺ᵠ in disguise, because logs are indices. Also loga a = 1 and loga 1 = 0, which lets you turn a plain number into a log of the same base.
How do you combine them step by step?
- Check the bases match. If not, change base first.
- Move every coefficient up as a power (law 3), before anything else.
- Add or subtract the logs using laws 1 and 2, working left to right.
- Write one log and simplify the number inside. If it is a power of the base, evaluate it.
- Turn any plain number into a log using n = n loga a = loga (aⁿ).
Worked example
Express 2 log₃ 6 − log₃ 4 + 1 as a single logarithm, then evaluate it.
Step 1, coefficients up: 2 log₃ 6 = log₃ 36.
Step 2, subtract: log₃ 36 − log₃ 4 = log₃ (36 ÷ 4) = log₃ 9.
Step 3, turn the 1 into a log: 1 = log₃ 3.
Step 4, add: log₃ 9 + log₃ 3 = log₃ (9 × 3) = log₃ 27.
Step 5, evaluate: 3³ = 27, so the value is 3.
Check without laws: log₃ 9 = 2, and 2 + 1 = 3. ✓ Both routes agree.
The mistake to watch for
A common slip is to multiply inside the log when the coefficient should become a power.
Mistaken working: 2 log₃ 6 = log₃ 12.
The student doubled the 6 instead of squaring it.
The correction is law 3: 2 log₃ 6 = log₃ 6² = log₃ 36.
A numerical test shows the difference. log₃ 36 is about 3.26, which is twice log₃ 6 (about 1.63).
But log₃ 12 is only about 2.26. Whenever a coefficient disagrees with your answer on a quick test, the coefficient has been misused.
A related trap is dividing logs: log₂ 8 ÷ log₂ 2 is 3 ÷ 1 = 3, which is not log₂ 4 (that equals 2). Subtracting logs gives a quotient inside, but dividing logs does not.
Check yourself
1. Write 3 lg 2 + lg 5 as a single logarithm.
Show answer
3 lg 2 = lg 8. Then lg 8 + lg 5 = lg 40. So the answer is lg 40.
2. Evaluate log₂ 48 − log₂ 3.
Show answer
Subtracting logs divides the numbers: log₂ (48 ÷ 3) = log₂ 16. Since 2⁴ = 16, the value is 4.
3. Simplify 2 log₅ 10 − log₅ 4 and give the value.
Show answer
2 log₅ 10 = log₅ 100. Then log₅ 100 − log₅ 4 = log₅ 25. Since 5² = 25, the value is 2.
Where this leads next
The next lesson uses a single log to solve an equation with a log-domain restriction, where you must also decide which answers are allowed. The non-calculator working trainer helps you rehearse exact arithmetic before moving to decimals.
If the laws still feel like separate tricks, a teacher in online one-to-one Additional Mathematics tuition can tie them back to index laws using your own working.