Exponent and Logarithm Calculator

What Is an Exponent?

An exponent tells you how many times to multiply a number, called the base, by itself. In the expression bⁿ, b is the base and n is the exponent (also called the power). So 2⁴ means "multiply 2 by itself 4 times": 2 × 2 × 2 × 2 = 16. Exponents are a compact shorthand for repeated multiplication — without them, writing out something like 2¹⁰ would mean spelling out ten separate multiplications every time.

A few special cases are worth memorizing: any nonzero number raised to the power of 0 equals 1 (b⁰ = 1), and any number raised to the power of 1 equals itself (b¹ = b). Exponents also grow extremely fast — this is called exponential growth, and it's why exponents show up everywhere from compound interest to population growth to computer science (doubling storage capacities, for example).

Negative and Fractional Exponents

Exponents aren't limited to positive whole numbers. Two other cases come up constantly:

Negative Exponents

A negative exponent means "take the reciprocal." Specifically, b&supminus;ⁿ = 1 / bⁿ. For example, 2&supminus;³ = 1 / 2³ = 1/8 = 0.125. The negative sign does not make the result negative — it flips the base into a fraction. This is especially useful in scientific notation, where a negative exponent on 10 represents a very small number (like 10&supminus;³ = 0.001).

Fractional Exponents

A fractional exponent represents a root. Specifically, b^(1/n) is the nth root of b. For example, 9^(1/2) is the square root of 9 (which is 3), and 8^(1/3) is the cube root of 8 (which is 2). More generally, b^(m/n) means "take the nth root of b, then raise it to the mth power" (or equivalently, raise b to the mth power first, then take the root — both give the same answer).

Negative Bases With Fractional Exponents

Things get trickier when the base is negative and the exponent is a fraction. A negative base raised to a fractional exponent often requires taking an even root of a negative number — for example, (−4)^(1/2) asks for the square root of −4, which has no real-number answer (it's a complex number, involving the imaginary unit i). This calculator sticks to real numbers, so when an exponent would require an even root of a negative base, it reports that the result is complex rather than showing an invalid number.

What Is a Logarithm?

A logarithm is the inverse operation of exponentiation. Where exponentiation asks "what do I get if I raise b to the power of n?", a logarithm asks the opposite question: "what power do I need to raise b to, in order to get x?" Formally, logₜ(x) = n means the exact same thing as bⁿ = x. For example, since 10³ = 1000, it's also true that log₁₀(1000) = 3.

Every logarithm has a base, just like every exponent has one, and the base must be positive and not equal to 1 (a base of 1 would give 1ⁿ = 1 for every exponent, so there'd be no unique answer to "what power gives x?" unless x itself is 1). The value you're taking the log of, x, must also be positive — there is no real power you can raise a positive base to that produces zero or a negative number.

Common Log vs. Natural Log

Two logarithm bases show up so often that they get their own shorthand notation:

• Common log (base 10) is written log(x) with no base shown, since 10 is assumed. It's the natural fit for anything built around powers of ten — the Richter scale for earthquake magnitude, the decibel scale for sound intensity, and pH in chemistry are all base-10 logarithmic scales. It's also the most intuitive base for mental estimation, since log₁₀(1000) = 3 just counts the zeros.
• Natural log (base e) is written ln(x), where e is Euler's number, approximately 2.71828. It shows up constantly in calculus, continuous compound interest, radioactive decay, and population growth models, because e is the unique base whose exponential function grows at a rate exactly equal to its own value at every point — a property that makes calculus involving exponentials dramatically simpler.

Both are just logarithms with a specific base baked in; everything you can do with logₜ(x) for any other base b works exactly the same way for these two, they simply come up often enough to earn dedicated names and buttons above.

Frequently Asked Questions

What does a negative exponent mean?

A negative exponent tells you to take the reciprocal of the positive-exponent version: b&supminus;ⁿ = 1 / bⁿ. It does not make the result negative. For example, 5&supminus;² = 1 / 5² = 1/25 = 0.04 — still a positive number, just a small fraction instead of a large one.

Why can't the base of a logarithm be 1 or a negative number?

A base of 1 fails because 1 raised to any power is always 1, so log₁(x) would have no unique answer for any x other than 1 itself — the question "what power of 1 gives me 5?" simply has no answer. A negative base fails for a different reason: raising a negative number to different exponents jumps back and forth between positive and negative (or produces complex numbers for non-integer exponents), so there's no consistent, well-behaved logarithm function that could be defined for it. Both restrictions exist to guarantee logₜ(x) always has exactly one real answer.

What is e and why does it matter for logarithms?

e (Euler's number) is an irrational constant approximately equal to 2.71828. It's the base of the natural logarithm (ln) because the exponential function eₓ has a uniquely elegant property: its rate of change at any point equals its own value at that point. This makes it the natural base for describing continuous growth or decay — compound interest calculated continuously, radioactive decay, and many population models are all most simply expressed using e, which is why ln shows up so often in science and calculus instead of log base 10 or any other base.

How do I calculate a logarithm by hand without a calculator?

For simple cases, ask "what power of the base gives this number?" For log₂(8), you're looking for the power of 2 that equals 8 — since 2³ = 8, log₂(8) = 3. For values that aren't clean powers, you can use the change-of-base formula: logₜ(x) = log(x) / log(b), using log or ln from any calculator that has those built in (this is exactly how this tool computes logarithms of any base internally, using Math.log(x) / Math.log(b)). Without any calculator at all, you can estimate by finding the two whole-number powers of the base that bracket your value and interpolating between them.

What's the relationship between exponents and logarithms?

They are inverse operations of each other, the same way addition and subtraction (or multiplication and division) undo one another. If bⁿ = x, then logₜ(x) = n — both statements say exactly the same thing, just solved for a different variable. This is why you can "verify" a logarithm result by raising the base to that result and checking that you get back your original value, which is exactly what the verification line under the Logarithm Calculator above shows.

Why does this calculator say some results are complex numbers instead of just showing a number?

Certain combinations — specifically a negative base raised to a non-integer exponent — don't have a real-number answer. For example, the square root of −9 isn't any real number, because no real number multiplied by itself produces a negative result. Mathematically, these cases do have an answer, but it involves the imaginary unit i (where i² = −1) and falls into the realm of complex numbers, which this calculator intentionally doesn't attempt to display, since real-world uses of this tool almost never call for a complex result.