What this calculator does
The logarithm laws are the rules that let a logarithm of a product, a quotient or a power be rewritten as a sum, a difference or a multiple of simpler logarithms. The product rule states log(ab) = log(a) + log(b), the quotient rule states log(a/b) = log(a) − log(b), and the power rule states log(aⁿ) = n·log(a).
This calculator takes two numbers, a base and a power, and computes both sides of all three logarithm laws at once, so you can see directly that they match rather than just being told the rule. It is a companion to a plain logarithm calculator: instead of evaluating one log, it demonstrates why the laws work on the specific numbers entered.
The formula
Each law is checked by computing both sides independently and showing they agree. For the product law, log_b(a×b) is computed directly and compared against log_b(a) + log_b(b). The quotient and power laws work the same way, computing the left side of the equation and the right side by the law, then displaying both.
| Term | Meaning |
|---|---|
| Product law | log_b(a×b) = log_b(a) + log_b(b). |
| Quotient law | log_b(a÷b) = log_b(a) − log_b(b). |
| Power law | log_b(aⁿ) = n × log_b(a). |
The inputs explained
| Field | What to enter |
|---|---|
| Number a | The first number, must be greater than zero. |
| Number b | The second number, must be greater than zero. |
| Base | The logarithm base, must be greater than zero and not equal to 1. |
| Power n (for the power law) | The power used for the power law only, applied to a. |
When to use it
Checking working on a logarithm problem
A student who has expanded log(ab) into log(a) + log(b) by hand can enter the same numbers here and confirm both sides genuinely match, before moving on to the next step of a proof or a problem set.
Understanding why the laws hold, not just memorising them
Seeing log_b(a×b) computed one way and log_b(a) + log_b(b) computed another way, landing on the same number, makes the identity concrete rather than abstract.
Simplifying an expression before solving an equation
Logarithmic equations are often solved by combining terms with these laws first. Testing the combination on sample numbers here checks that the simplification step was applied correctly.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
The three laws at base 10, across a range of number pairs
A fixed second number of 5, base 10, with the first number varying.
| a | Product law: log(a×b) | Quotient law: log(a÷b) | Power law: log(aⁿ) |
|---|---|---|---|
| 2 | 1.000 = log(a) + log(b) = 1.0000 | -0.3979 = log(a) − log(b) = -0.3979 | 0.6021 = n × log(a) = 0.6021 |
| 5 | 1.398 = log(a) + log(b) = 1.398 | 0 = log(a) − log(b) = 0 | 1.398 = n × log(a) = 1.398 |
| 10 | 1.699 = log(a) + log(b) = 1.699 | 0.3010 = log(a) − log(b) = 0.3010 | 2.000 = n × log(a) = 2.000 |
| 20 | 2.000 = log(a) + log(b) = 2.000 | 0.6021 = log(a) − log(b) = 0.6021 | 2.602 = n × log(a) = 2.602 |
| 50 | 2.398 = log(a) + log(b) = 2.398 | 1.000 = log(a) − log(b) = 1.0000 | 3.398 = n × log(a) = 3.398 |
| 100 | 2.699 = log(a) + log(b) = 2.699 | 1.301 = log(a) − log(b) = 1.301 | 4.000 = n × log(a) = 4.000 |
How the power law scales with the exponent
A fixed a of 2 at base 2, so log_2(2) = 1, across a range of exponents n.
| Power n | Power law: log(aⁿ) |
|---|---|
| 1 | 1.000 = n × log(a) = 1.000 |
| 2 | 2.000 = n × log(a) = 2.000 |
| 3 | 3.000 = n × log(a) = 3.000 |
| 4 | 4.000 = n × log(a) = 4.000 |
| 5 | 5.000 = n × log(a) = 5.000 |
| 10 | 10.000 = n × log(a) = 10.000 |
Questions
What are the three main logarithm laws?
The product law turns a log of a product into a sum of logs: log(ab) = log(a) + log(b). The quotient law turns a log of a quotient into a difference: log(a/b) = log(a) − log(b). The power law pulls an exponent out the front as a multiplier: log(aⁿ) = n·log(a).
Do the logarithm laws work in any base?
Yes. All three laws hold for any valid logarithm base, as long as the base is positive and not equal to 1, and the numbers being logged are positive. Changing the base changes the numeric values but not whether the laws hold.
Why does the power law let you bring the exponent to the front?
Because raising a number to a power n is the same as multiplying it by itself n times, and the product law applied repeatedly to that repeated multiplication collapses into n separate additions of the same log, which is just n times that log.
How is this different from a normal logarithm calculator?
A plain logarithm calculator evaluates log_b(x) for one number. This tool instead takes two numbers and shows both sides of the product, quotient and power laws computed independently, so the identity itself is visible rather than assumed.
To evaluate a single logarithm at any base, see the logarithm calculator. For the reverse operation, raising a base to a power, see the exponents and roots calculator.