Powers, Roots & Logs Calculators

One family of inverse operations

Powers, roots and logarithms are not three separate topics; they are three ways of looking at the same equation. If 2 raised to the power 5 equals 32, then the fifth root of 32 is 2, and the logarithm of 32 to base 2 is 5. One statement, three questions: given the base and exponent, find the result; given the result and exponent, find the base; given the result and base, find the exponent. The five free calculators on this shelf cover each direction, plus the two places where exponents dominate everyday practical work: writing very large or very small numbers in scientific notation and tracking exponential decay with the Half-Life Calculator.

Because the tools share one underlying idea, they work well in sequence. A problem that starts in the Exponent Calculator often ends in the Log Calculator, because solving for an unknown exponent is exactly what a logarithm does. Keep that inverse relationship in mind and half the shelf explains itself.

Exponents: from repeated multiplication to fractional powers

An exponent begins life as shorthand for repeated multiplication: 3 to the power 4 means 3 × 3 × 3 × 3, which is 81. The Exponent Calculator handles that case with the multiplication chain written out, but its real value is in the cases schools rush past. A negative exponent means a reciprocal, so 2 to the power of minus 3 is 1 divided by 2 cubed, which is 1/8. A zero exponent gives 1 for any non-zero base. And a fractional exponent combines a root with a power: raising a number to the power 2/3 means squaring it and then taking the cube root, in either order. The calculator shows whichever of these routes applies to your input, with exact whole-number answers printed in full rather than collapsed into rounded notation.

The exponent rules follow directly from the repeated-multiplication picture. Multiplying two powers of the same base adds the exponents; dividing subtracts them; raising a power to a power multiplies them. When a result grows past what a display can sensibly show, the Big Number Calculator takes over and computes the whole thing exactly, however many digits it runs to.

Roots and radicals, simplified properly

A root asks the exponent question in reverse: which number, raised to this power, gives that result? The Root Calculator answers it two ways at once. It gives the precise decimal value, and it simplifies the root to its exact radical form using prime factors, which is the version maths courses actually ask for. The square root of 72, for example, is not just 8.485 and change; it is 6 times the square root of 2, because 72 factors into 36 × 2 and the 36 comes out of the radical as 6.

Why exact form matters

Decimal answers are almost always truncated, so errors creep in when you square them again or feed them into later steps. Exact radical form carries no error at all. The calculator also runs a perfect-power check, so you know immediately whether a root comes out to a whole number. Roots turn up constantly elsewhere on the site: the Pythagorean Theorem Calculator leaves its answers as exact surds for the same reason, and the Quadratic Formula Calculator does the same when the discriminant is not a perfect square.

Logarithms: solving for the exponent

A logarithm answers the question the other two tools cannot: to what power must I raise this base to get that number? The Log Calculator evaluates a logarithm in any base you choose, and it also runs the question in its natural phrasing, base to the power of what gives x, which is often how the problem actually arrives in your head.

Three bases cover nearly everything. Base 10, the common logarithm, underlies decibels and pH. Base 2 runs through computing, where it counts doublings and bits. And base e, the natural logarithm, is the native language of continuous growth and decay. For any other base, the calculator applies the change-of-base rule and shows it: the log of x to base b equals the natural log of x divided by the natural log of b. That one identity is why a calculator with only an ln key can still evaluate any logarithm ever written.

The rules, briefly

Logarithms convert multiplication into addition: the log of a product is the sum of the logs, the log of a quotient is the difference, and the log of a power pulls the exponent out front as a multiplier. That last rule is the workhorse. It is how you solve for time in compound growth, and it is exactly the move the Compound Interest Calculator relies on when it works out how long money takes to double.

Scientific notation: writing awkward numbers honestly

Once numbers pick up more than a handful of zeros, plain decimal notation stops communicating. Scientific notation writes any number as a value between 1 and 10 multiplied by a power of 10, so 4,830,000 becomes 4.83 × 10 to the power 6. The Scientific Notation Calculator converts any number into standard, scientific, E and engineering notation at once. Engineering notation restricts the exponent to multiples of 3 so the number lines up with unit prefixes such as kilo, mega and micro, which is why engineers prefer it.

The calculator also does arithmetic directly in scientific notation, with the rules worked out on screen. Multiplication multiplies the leading values and adds the exponents; division subtracts them; addition and subtraction first rewrite both numbers to a matching exponent, then combine. Whenever a result's leading value drifts outside the 1-to-10 range, the tool shows the renormalising step that puts it back. If you are juggling significant figures at the same time, the Rounding Calculator is the companion tool, since scientific notation is the one format that makes significant figures completely unambiguous.

Half-life: exponents doing real work

The Half-Life Calculator is where this shelf leaves the classroom. Radioactive decay, and plenty of other processes, follow the decay law: the amount remaining equals the starting amount times one half raised to the power of elapsed time divided by the half-life. The calculator solves that law for whichever quantity you are missing: the amount left, the starting amount, the elapsed time or the half-life itself. Solving for time or half-life requires taking a logarithm, which is the inverse relationship from the top of this page put to practical use.

Alongside every answer it reports the decay constant and the mean lifetime, the two other standard ways of describing the same decay rate. The decay constant equals the natural log of 2 divided by the half-life, and the mean lifetime is its reciprocal. The same mathematics powers carbon dating, drug elimination in the body and the fade of anything that loses a fixed proportion per unit time. It is also, with the sign flipped, the mathematics of compound growth, which is why the working here should look familiar if you have spent time on the Saving & Investing shelf.

Frequently asked questions

What does a negative exponent actually mean?

A reciprocal, not a negative number. 10 to the power of minus 2 is 1/100, which is 0.01, a small positive number. A negative result from a power needs a negative base, such as minus 2 cubed giving minus 8. The Exponent Calculator shows the reciprocal step whenever it applies.

What is the difference between ln and log?

ln is the logarithm to base e, roughly 2.71828. On its own, log usually means base 10 in schoolwork and engineering, though some fields and programming languages use log for the natural logarithm. The Log Calculator lets you set the base explicitly, which removes the ambiguity.

Can I take the square root of a negative number?

Not within the real numbers; no real number squares to a negative. Odd roots are fine, though: the cube root of minus 8 is minus 2. The Root Calculator follows exactly this rule.

Why does my calculator show 5e+9 instead of a normal number?

That is E notation, a compact form of scientific notation meaning 5 × 10 to the power 9, or 5,000,000,000. Paste it into the Scientific Notation Calculator to see it in every standard format, or use the Scientific Calculator when it appears mid-expression. Both tools are free and show the conversion step by step.