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Properties of Logarithms Calculator

Online logarithm properties calculator: convert multiplication and division into addition and subtraction of logs, with automatic expansion and evaluation.

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log_a(MN)=log_a M+log_a N; log_a M^n=n log_a M
a>0,a≠1
Positive
Positive

📖 Tutorial | Log Properties

1. Definition
Under a>0,a≠1 and M,N>0, logarithms convert multiplication, division and exponentiation into addition, subtraction and multiplication.
2. Formula & Notation
log_a(MN)Product
log_a(M/N)Quotient
log_a M^nPower
3. Properties & Laws
  • log_a(MN)=log_a M+log_a N;
  • log_a(M/N)=log_a M−log_a N;
  • log_a M^n=n log_a M;
  • log_a 1=0, log_a a=1;
  • Change of base: ln N/ln a.
4. Steps
  1. Enter the base a and numbers M,N;
  2. Click "Calculate" to expand;
  3. Or click "Load Example".
5. Worked Examples
Example: log₂(8×16)=log₂8+log₂16=3+4=7.
6. Common Pitfalls
Check arguments are positive first;
log_a(M±N) cannot be split;
Power gives n log_a M, not (log_a M)^n.

❓ FAQ | Log Properties

Product rule?
Sum of the logs.
Quotient rule?
Difference of the logs.
Power rule?
Bring n down as a multiplier.
Conditions?
a>0,a≠1, positive arguments.
log_a(M+N)?
No such rule.
Why use them?
They reduce operations one level, simplifying work.