Expanding Logarithms Calculator
Expand a logarithm of products, quotients, and powers into separate terms. The result labels every rule and keeps the domain restriction visible.
Written and checked by TheToolNet Editorial Team · Last reviewed: September 19, 2026
Expand a logarithmic expression
Use * for multiplication, one / for the main quotient, ^ for powers, and sqrt(y) for a square root.
Logarithm expansion rules
Use the quotient rule before the product rule so the sign of every denominator term remains clear. Then use the power rule. Expansion changes form, not value, provided the original expression and all required logarithm arguments are valid in the real numbers.
| Pattern | Expanded form |
|---|---|
| logb(MN) | logbM + logbN |
| logb(M/N) | logbM − logbN |
| logb(Mp) | p logbM |
Worked example
For log10(x²y/z³), first split the quotient: log(x²y) − log(z³). Split the product: log(x²) + log(y) − log(z³). Move powers: 2log(x) + log(y) − 3log(z).
Frequently asked questions
What bases are supported?
Use a positive numerical base other than 1, e, or a symbolic base such as b.
Why does the domain matter?
A real logarithm needs a positive argument. Algebraically similar expressions can have different apparent domains if absolute-value conditions are omitted.
Does this solve logarithmic equations?
Use the separate logarithm equation solver for equations.