Math
Exponent and root calculator
Powers, roots and logarithms are the same relationship read three ways: which number, raised to which power, gives which result.
The three are one relationship
If 25 = 32, then the fifth root of 32 is 2, and log base 2 of 32 is 5. Each operation solves for a different unknown in the same equation. Powers find the result, roots find the base, logarithms find the exponent.
The exponent rules, and why they hold
| Rule | Reason |
|---|---|
| xa × xb = xa+b | The factors are simply counted together |
| xa ÷ xb = xa−b | Cancelling removes b copies |
| (xa)b = xab | b groups of a factors |
| x0 = 1 | Follows from xa ÷ xa |
| x−a = 1⁄xa | Extends the division rule below zero |
| x1/n = n√x | Because (x1/n)n = x1 |
Each of the last three is a definition chosen so the first three keep working. That is why zero and negative exponents behave the way they do — they are the only values that preserve consistency.
Why logarithms compress
A logarithm turns multiplication into addition, which is why they were invented — Napier's tables in 1614 let astronomers replace laborious multiplication with lookup and addition, and slide rules mechanised the same trick for three centuries afterwards.
The same property makes log scales useful for data spanning many orders of magnitude. The Richter scale, decibels, pH and stellar magnitude are all logarithmic, which is why a magnitude 6 earthquake releases roughly 32 times the energy of a magnitude 5, not 20% more.
Three bases worth knowing
- Base 10 — written log. Counts digits, and underlies scientific notation and the decibel.
- Base e — written ln, where e ≈ 2.71828. Appears wherever growth is proportional to current size: compound interest, radioactive decay, population models.
- Base 2 — written log₂. Counts binary digits and halvings, so it dominates computer science and algorithm analysis.
Any base converts to any other by dividing: logb(x) = ln(x) ÷ ln(b). Calculators that only offer ln and log can therefore handle every base.
Common questions
Why is any number to the power of zero equal to one?
Because x^a ÷ x^a is obviously 1, and the division rule says it equals x^(a−a) = x^0. Defining x^0 as 1 is the only choice that keeps the exponent rules consistent. The one exception is 0^0, which is left undefined in analysis, though it is often treated as 1 in combinatorics.
Can you take the square root of a negative number?
Not within the real numbers, since any real number squared is positive. Complex numbers extend the system by defining i as the square root of −1, which makes every root available. Odd roots of negatives are real, though: the cube root of −8 is −2.
What is e and why does it matter?
It is the value that makes the exponential function its own derivative, which means it describes any process whose rate of change is proportional to its current size. It arises naturally as the limit of (1 + 1/n)^n as n grows — the value continuously compounded interest converges to.
How do I calculate a logarithm in a base my calculator lacks?
Use change of base: divide the natural log of the number by the natural log of the base. For log base 7 of 343, compute ln(343)/ln(7) = 3.