The laws of logarithms turn multiplication into addition: logₐ(xy) = logₐ x + logₐ y, logₐ(x/y) = logₐ x − logₐ y, and logₐ(xᵏ) = k logₐ x. They let you combine several logarithms into one, split one into several, evaluate expressions exactly, and write one logarithm in terms of others that you are given.
The laws of logarithms
- Product law: logₐ(xy) = logₐ x + logₐ y
- Quotient law: logₐ(x/y) = logₐ x − logₐ y
- Power law: logₐ(xᵏ) = k logₐ x
- logₐ √x = ½ logₐ x and logₐ(1/x) = −logₐ x
The laws only work for logs with the same base. logₐ(x + y) cannot be split.
The product and quotient laws
Multiplying inside a logarithm becomes adding outside, and dividing becomes subtracting. These come directly from the index laws: multiplying powers adds the indices, and dividing subtracts them.
The laws only combine logarithms with the same base, and there is no law for log(x + y). Treating log(x + y) as log x + log y is a classic error.
The power law
A power inside a logarithm comes to the front as a multiplier: log(x³) = 3 log x. Two useful special cases follow. A square root is a power of ½, so log √x = ½ log x. A reciprocal is a power of −1, so log(1/x) = −log x.
Combining terms into a single logarithm
To collapse an expression into one logarithm, use the power law first to move every coefficient inside as a power, then use the product and quotient laws to combine. Positive terms end up on the top of the fraction and negative terms on the bottom. A plain number can be written as a log: for base 10, 1 = log 10 and 2 = log 100.
Splitting a logarithm into separate terms
Run the laws backwards to expand a single logarithm. Everything on the top of a fraction becomes a positive term, everything on the bottom a negative term, and every power comes down as a multiplier. This also works with numbers: factorise them into primes first, so log 12 = log(2² × 3) = 2 log 2 + log 3.
Exact values and “in terms of” questions
Chain the laws to reach an exact value: combine everything into one logarithm, then evaluate it. In “in terms of p and q” questions, you are given values such as logₐ 2 = p; factorise the target number into those primes and split it, so logₐ 20 = logₐ(2² × 5) = 2p + logₐ 5.
Worked examples
Example 1. Write log 4 + log 25 as a single logarithm and hence evaluate it.
- Product law: log(4 × 25) = log 100.
- log 100 = log 10² = 2.
Answer: 2.
Marks: M1 for log(4 × 25), A1 for 2.
Example 2. Write 2 log₅ 3 as a single logarithm.
- Power law in reverse: 2 log₅ 3 = log₅ 3².
Answer: log₅ 9.
Marks: A1 for log₅ 9.
Example 3. Write 2 log x + 3 log y − log z as a single logarithm.
- Power law first: log x² + log y³ − log z.
- Add means multiply; subtract means divide.
Answer: log(x²y³ / z).
Marks: M1 for moving the coefficients, A1 for the single log.
Example 4. Write logₐ(x³/y) as separate logarithms.
- Quotient law: logₐ x³ − logₐ y.
- Power law: 3 logₐ x − logₐ y.
Answer: 3 logₐ x − logₐ y.
Marks: M1 for the quotient law, A1 for the answer.
Example 5. Evaluate 2 log₃ 6 − log₃ 4.
- Power law: log₃ 36 − log₃ 4.
- Quotient law: log₃(36/4) = log₃ 9.
- 3² = 9.
Answer: 2.
Marks: M1 for log₃(36/4), A1 for 2.
Common mistakes
- Splitting log(x + y). There is no law for a sum inside a log.
- Combining before using the power law. Move every coefficient inside as a power first.
- Writing log x / log y as log(x/y). log(x/y) = log x − log y; a quotient of logs is something else.
- Mixing bases. The laws only combine logs with the same base.
- Getting the sign wrong for a reciprocal. log(1/x) = −log x.
How the laws of logarithms are examined
The log laws are tested heavily in Pure 2 (WMA12) of Edexcel International A-Level, and in Year 1 of Edexcel A-Level (9MA0). Questions ask you to:
- combine several logs into a single logarithm
- expand a single log into separate terms
- show that an expression has an exact value
- write a logarithm in terms of given letters, such as p and q
Work through the exam questions on exponentials and logarithms and mark your answers against the schemes, then try the Pure 2 exponentials and logarithms worksheet. The exponentials and logarithms revision engine gives unlimited extra practice.
The laws are the main tool in solving equations with logarithms, and they work for ln as well, in natural logarithms.
Frequently asked questions
What are the three laws of logarithms?
The product law, log(xy) = log x + log y; the quotient law, log(x/y) = log x − log y; and the power law, log(xᵏ) = k log x.
Is log(x + y) equal to log x + log y?
No. There is no law that splits the logarithm of a sum.
Why do the log laws work?
Because logs are powers. Multiplying numbers adds their powers, dividing subtracts them, and raising to a power multiplies them.