A logarithm is a power, read backwards
If an exponent answers the question what do I get when I raise this base to this power, a logarithm answers the same question turned around: to what power must I raise this base to get this number? Since 210 = 1024, the logarithm of 1024 in base 2 is 10. Every logarithm statement and every power statement are two spellings of one fact, and this calculator keeps that check visible — the breakdown raises the base to your result and shows your starting number coming back, so you never have to take the answer on trust.
Three bases dominate practice, and this page reports all three for your number alongside the base you chose. Base 10, written log, is the scale of everyday magnitudes: pH, decibels, the Richter scale and orders of magnitude all count in tens. Base e, written ln and called the natural logarithm, is the base of growth and decay in calculus, finance and physics, where e is about 2.71828. Base 2, written log2, counts doublings and bits in computing: how many binary digits a number needs, how many halvings a search takes. Seeing all three side by side makes their family resemblance obvious — they differ only by a constant multiplier, the change-of-base ratio.
The change-of-base formula
Most calculators only offer ln and log base 10, so any other base is reached with this formula: divide the logarithm of the number by the logarithm of the base, using any one consistent base for both. It works because a logarithm measures a ratio of growth, and changing the ruler you measure with rescales every reading by the same factor. This page does that division for you with the natural log, which is why the working panel shows ln(x) divided by ln(base).
The rules logarithms inherit from powers
Because logarithms are exponents, the power rules reappear in mirror form. The log of a product is the sum of the logs — multiplication downstairs becomes addition upstairs, the trick that made slide rules and log tables the fast calculators of three centuries. The log of a quotient is a difference, the log of a power brings the exponent down in front, and the log of 1 is always 0 in every base, because any base to the power 0 is 1. Two honest limits follow from the definition: there is no logarithm of zero or of a negative number in real arithmetic, since no power of a positive base ever reaches them, and base 1 is forbidden because 1 to any power stays 1 and undoes nothing. This page says so plainly rather than printing a misleading number.
Where logarithms earn their keep
Logarithms turn multiplicative worlds into additive ones, and human senses are built that way: we hear loudness and see brightness roughly logarithmically, which is why decibels and camera stops feel evenly spaced while the underlying energy multiplies. In data work, a quantity that grows by percentages — an investment, an epidemic curve, a follower count — becomes a straight line on a logarithmic scale, and deviations from the trend become visible at a glance. And in computing, log base 2 is the honest measure of search: a well-organised index of a billion entries needs only about 30 halvings to find any entry, a fact that still surprises people meeting it for the first time.
Worked example
Log base 2 of 1024 is 10, because 2 to the power 10 is 1024. For the same number, log base 10 is about 3.0103, the natural log is about 6.9315, and log base 2 is exactly 10.
How to use this calculator
- Enter the number x — it must be greater than zero.
- Enter the base — greater than zero and not equal to 1; 2, 10 and e are the common choices.
- Read your base result in the green panel.
- Compare log base 10, natural log and log base 2 for the same number in the breakdown.
- Use the check line: the base raised to the result should return your number.
Antilogarithms: going back the other way
The inverse of a logarithm is simply a power: if log base 2 of x is 10, then x is 2 to the power 10. Older textbooks called this the antilog, and log tables printed antilog tables beside the log tables so engineers could travel in both directions. Today the Exponent Calculator covers the return journey, and the Scientific Calculator offers ln and log inside longer expressions. A useful sanity habit links the two directions forever: whatever a log gives you, raising the base to it must hand your number back — and if it does not, a base has been mixed up somewhere.