What is a natural logarithm?

It is the area under the curve y = 1/t from t = 1 up to t = x. That area is what the symbol ln x names, and it is a complete definition on its own — nothing about bases or powers is needed to state it. The more familiar description, that ln is the logarithm to base e, is a consequence: e turns out to be the number whose area is exactly 1. Both descriptions pick out the same function, but only the first explains why anyone would care about this particular one.

Why base e rather than base 10?

Because e is the only base for which the slope of the logarithm at x is exactly 1/x, with no stray constant multiplying it. Every other base introduces a factor that then follows you through every calculation. Base 10 is genuinely convenient for counting digits and base 2 for repeated halving, and neither is wrong — but as soon as rates of change are involved, any other base leaves a constant lying around, and e is the choice that makes it vanish.

It is roughly 2.718281828459045, and it is defined by the property this page draws: it is the number you have to reach for the area under 1/t, starting from 1, to equal exactly 1. Equivalently it is what (1 + 1/n)ⁿ approaches as n grows, which is the compound-interest version of the same fact — interest compounded ever more frequently at 100 per cent approaches a factor of e. It is irrational and transcendental, so it has no exact decimal or fractional form, only better and better approximations.

When should I use ln rather than log?

Use ln whenever a rate of change is involved: continuous growth or decay, half-lives, compound interest taken to its limit, anything with a differential equation behind it. Use log base 10 when you want to count digits or read a decibel or pH scale, and log base 2 when you are halving or doubling. Notation is a trap here — in most mathematical writing and in nearly every programming language, plain "log" means the natural one, while on a pocket calculator and in engineering it usually means base 10.

Why is there no logarithm of zero or a negative number?

Because no real power of e produces one. e is positive and stays positive however it is raised, so nothing it can reach is ever zero or below. Seen as an area, the same thing appears differently: as x falls toward zero the region under 1/t grows without limit in the negative direction, so the value runs off to minus infinity rather than settling anywhere. Negative numbers do have logarithms once complex numbers are allowed — ln(−a) = ln a + iπ — but the answer stops being a single real value.

How do I convert between ln and other bases?

Divide by the natural log of the base you want: log_b x = ln x ÷ ln b. So log₁₀ x is ln x ÷ 2.302585092994046 and log₂ x is ln x ÷ 0.6931471805599453. Because the divisor does not depend on x, every logarithm is the same curve with the vertical axis rescaled, which is why they all cross zero at x = 1 and never cross each other anywhere else. Worth knowing for accuracy as well as tidiness: for base 2 and base 10 specifically, a dedicated routine is more accurate than doing the division yourself, and this page uses those.

Why does ln of a number just above 1 come out wrong?

The logarithm is not at fault — the addition before it is. To compute ln(1 + d) a calculator first forms 1 + d, and a double cannot hold both a 1 and a very small d, so the small part is rounded away before the logarithm is reached. Measured against a 60-digit reference, the naive route is off by about 6×10⁻⁹ at d = 10⁻⁸, by 9×10⁻⁵ at 10⁻¹², by 11 per cent at 10⁻¹⁵, and at 10⁻¹⁶ it returns exactly zero. The fix is log1p, which computes the same quantity without ever forming the sum, and this page offers a d field so you can use it.

How accurate are the values on this page?

Checked against a 60-digit fixed-point series computed independently of the browser, the worst relative error across 6,001 values spanning e⁻³⁵ to e³⁵ was 1.4×10⁻¹⁶, which is under one unit in the last place of a double. Raising e back to each answer returned the original number bit for bit on every one of 4,000 further tests. The page reports that round trip beside every result, in units in the last place, so you are not asked to take any of this on trust.

Where are natural logs actually used?

Anywhere something changes at a rate proportional to how much of it there is. Radioactive half-lives, continuously compounded interest, population growth, the discharge of a capacitor and the cooling of a hot object are all the same equation with different names. Logs are also how quantities spanning many orders of magnitude are made comparable — pH, decibels, earthquake magnitudes, the Richter scale — and how enormous products are computed at all, by adding logs instead of multiplying numbers too large to hold.

Natural Logarithm Calculator

Take the natural logarithm of any positive number, to the full precision a computer can hold, including the one case where almost every calculator quietly gets it wrong. The usual definition — "log to the base e" — has the story backwards, and it makes e look like an arbitrary constant somebody pic...

ln x = the area under y = 1/t from t = 1 to t = x

e is not picked first and the log built around it. e is the value of x at which that area first reaches 1 — the base comes out of the definition rather than going into it.

THE NUMBER

xBASE

x is very close to 1. Enter d instead, where x = 1 + d

LN x

0.693147180559945

e raised to that gives 2 — exactly x again, to the last bit.

LOG₁₀ x

0.301029996

LOG₂ x

1.000000000

LOG BASE 10

0.301029996

Quantity	Value	What it is
ln x	0.693147180559945	the area under 1/t from 1 to x
log₁₀ x	0.301029995663981	ln x divided by ln 10
log₂ x	1	ln x divided by ln 2
e^(ln x)	2	x again, exactly
d/dx at x	0.5	the slope of ln there, which is 1/x
ln(x²)	1.38629436111989	exactly twice ln x, since powers come down in front

Why this one is the natural logarithm. Every logarithm is a constant multiple of every other, so as curves they are all the same shape — ln x divided by log₁₀ x is 2.302585092994046 whatever x you choose. What singles this one out is that its slope at x is exactly 1/x, with no leftover constant in front. Base 10 is convenient for counting digits and base 2 for halving, but only base e makes the calculus come out clean, which is why it turns up wherever something grows at a rate proportional to its own size.

Multiplication becomes addition, which is the entire point. ln of 2 squared is 1.38629436112, exactly twice the figure above — no multiplication was performed to get it. That is what logarithms were invented for, and it is still why they are used: numbers far too large to hold can be multiplied by adding their logs, growth at a constant percentage becomes a straight line, and quantities spanning many orders of magnitude become comparable.

THE AREA

Drag the slider and watch the area accumulate. The number it reaches is the logarithm — not an approximation of it, but what it means. Sweeping past 1 in the other direction accumulates negative area, which is why numbers below 1 have negative logs.

Live calculation · updates as you type

Created with❤️byeaglecalculator.com

HOW TO USE

1
Type any positive number into x. The natural log appears immediately, along with log base 10, log base 2, and a log in whatever base you put in the base box.
2
If your number is very close to 1, tick the box and enter d instead, where x = 1 + d. This is not a convenience — it is the only way to get the right answer, and the page shows you what the ordinary calculation would have returned.
3
Check the round-trip line under the answer. It raises e back to the logarithm and reports how far that lands from your x, measured in units in the last place, so you can see the arithmetic close rather than take it on trust.
4
Drag the slider to sweep the area out from 1 to x. That accumulating area is not a picture of the logarithm, it is the definition of it — which is why numbers below 1 come out negative, the region being swept the other way.

REFERENCE FORMULAS

Rule	Formula	What it is for
Definition	ln x = ∫₁ˣ dt / t	The area under 1/t from 1 to x. Everything else follows from this one line.
What e is	ln e = 1	e is the x at which that area first equals 1. The base is a consequence, not a choice.
Product	ln(ab) = ln a + ln b	Multiplication becomes addition. The reason logarithms were invented at all.
Quotient	ln(a/b) = ln a − ln b	And division becomes subtraction, which is the same statement.
Power	ln(aⁿ) = n · ln a	Exponents come down in front. This is what makes log scales linear in growth rate.
Change of base	logᵇ x = ln x / ln b	Every logarithm is ln scaled by a constant, so all of them have the same shape.
Derivative	d/dx ln x = 1/x	The only base for which this is true with no stray constant. That is "natural".
Near one	ln(1 + d) ≈ d − d²/2 + d³/3	For small d, ln(1 + d) is very nearly d itself — and log1p is how to compute it.
Doubling time	t = ln 2 / r	Continuous growth at rate r doubles in this time. ln 2 = 0.6931471805599453.
Inverse	e^(ln x) = x	Checked on this page every time you type, and reported in units in the last place.

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Last updated: August 3, 2026 · Checked against a 60-digit fixed-point series · log1p used for values near 1 · every answer round-tripped through e^x