Expanding logarithms breaks a complicated logarithm into simple pieces. It is used to differentiate products (logarithmic differentiation), to linearise data and to simplify expressions before solving.
How to use the Expanding Logarithms Calculator
- Expression
- a
- b
- Exponent k
- Logarithm base
- Custom base
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
Choose the form, enter the numbers and the base.
Formula
log(a·b) = log a + log b, log(a/b) = log a − log b, log(aᵏ) = k·log a
Worked example
Inputs — Expression: Product log(a·b); a: 8; b: 4; Logarithm base: 10 (common log).
Results — Expanded form = log₁₀(8) + log₁₀(4); Original = log₁₀(8 · 4); Value = 1.50514997832; Rule = Product rule: log(a·b) = log a + log b.
Key terms
- Power rule
- log(aᵏ) = k log a.
Frequently asked questions
How do I expand log(x²y)?
2 log x + log y.
Is log(a + b) = log a + log b?
No – there is no rule for the log of a sum.
Why expand logarithms?
To simplify derivatives, equations and data analysis.
Do the rules work in every base?
Yes.