Calculate a number raised to a power (exponent), or find the logarithm of a number in any base.
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Exponents and logarithms are two complementary operations in mathematics: an exponent means multiplying a number by itself a certain number of times, while a logarithm is the reverse operation — it answers the question "to what power must the base be raised to get a given number?" This calculator offers two separate tools: the first calculates any number raised to any power, and the second calculates the logarithm of a number to any base you choose, using the change-of-base formula log_b(x) = ln(x)/ln(b). Both operations are used in physics, economics, and computer science. Because exponents and logarithms are true inverses of each other, raising a base to the logarithm of a number (using that same base) always returns the original number exactly — a relationship that is useful both for checking a logarithm calculation by hand and for understanding why the two functions are so often taught side by side.
Exponents and logarithms are inverse operations of each other in exactly the same sense that addition and subtraction, or multiplication and division, are inverses. Raising 10 to the power of 3 gives 1,000; asking 'what power of 10 gives 1,000' — the logarithm base 10 of 1,000 — gives back 3. Understanding this reciprocal relationship makes both operations far more intuitive than memorizing them as two unrelated topics.
Exponents beyond whole-number powers extend the concept further: a fractional exponent represents a root (x^(1/2) is the same operation as the square root of x), and a negative exponent represents a reciprocal (x⁻¹ equals 1/x). These extensions are not arbitrary conventions — they follow directly from the rules of exponents applied consistently, which is exactly why x^(1/2) squared must equal x to the first power, matching the definition of a square root.
The change-of-base formula, log_b(x) = ln(x) / ln(b), solves a practical problem: most calculators and programming languages only provide a natural logarithm (ln, base e) or common logarithm (log, base 10) function directly, but real problems frequently require a logarithm in some other base — base 2 in computer science, for example. The change-of-base formula lets any logarithm be computed from whichever base function is actually available, which is precisely the method this calculator uses internally.
Base-10 logarithms appear throughout science in scales designed to compress an enormous range of values into a manageable number: the Richter scale for earthquake magnitude, the decibel scale for sound intensity, and the pH scale for acidity are all logarithmic, meaning each whole-number increase represents a tenfold change in the underlying quantity. A magnitude 6 earthquake releases roughly ten times the energy of a magnitude 5 earthquake — a fact that is only clear once the logarithmic nature of the scale is understood.
Base-2 logarithms are foundational to computer science, since they answer 'how many bits are needed to represent this many distinct values' — the logarithm base 2 of 256 is 8, exactly matching the fact that a single byte (8 bits) can represent 256 distinct values. Exponential growth, meanwhile, governs compound interest in finance, population growth in biology, and radioactive decay in physics (via its inverse, exponential decay), making the pair of operations covered by this calculator relevant across an unusually wide range of academic and professional fields.
Yes, the exponent field accepts negative and decimal values, such as 2^-1 = 0.5 or 4^0.5 = 2.
10, but you can enter any positive base other than 1, including e (2.71828) for natural log.
Because log base 1 is mathematically undefined — 1 raised to any power always equals 1, so it can never reach other values.