How this instrument works
Natural log answers one question: to what power must e be raised to produce x? Write it ln(x), and it is the inverse of the exponential function eˣ — feed ln(x) back into e^(·) and x reappears exactly. The 'natural' in the name is not decoration; e ≈ 2.718281828… is the unique base whose exponential function equals its own derivative, so calculus statements about continuous growth, interest compounded infinitely often, or radioactive decay reduce most cleanly when written in base e rather than base 10 or base 2.
The domain is one-sided by construction: ln(x) is defined only for x > 0, because no real power of the positive base e ever produces zero or a negative number — the curve y = eˣ never touches the x-axis, so its mirror image, y = ln(x), never crosses into x ≤ 0. As x approaches zero, ln(x) plunges toward negative infinity; as x grows without bound, ln(x) keeps climbing too, just far more slowly than x itself — ln of one million is only about 13.8.
One identity is worth knowing by heart: ln(a·b) = ln(a) + ln(b), so multiplication under the natural log becomes addition, the same trick that made slide rules work. It is also why ln(10) turns up everywhere as a conversion constant: multiply any base-10 logarithm by ln(10) ≈ 2.302585 and the result is the natural logarithm of that same number — a bridge used constantly when switching between the two systems, and the exact figure this sheet returns when x is set to 10.
- Enter your positive number into the x field — this is the value you want the natural log of.
- Read the ln(x) field for the answer: the exponent e must be raised to in order to reproduce x.
- Check it against the formula box if you like — raising e to that exponent should return your original x.
- Need a base-10 log instead? Multiply the ln(x) reading by 0.4342944819, which is 1 ⁄ ln(10).
Worked example — growing tenfold at a continuous rate
A population growing continuously at a 100% annual rate, following dN/dt = N, needs t = ln(x) years to multiply by any factor x. Entering x = 10 into the x field returns result = ln(10) = 2.302585092994046 years — call it 2.3026 years, roughly 2 years and 4 months, for the population to grow tenfold along a pure eᵗ curve.
The figure checks itself both ways. Raising e to the power 2.302585092994046 returns 10 exactly, since eˡⁿ⁽ˣ⁾ = x is simply the definition being inverted. The instrument carries the value to full double precision — 2.302585092994046 — so the digits displayed are the digits computed, with nothing trimmed or estimated along the way.
Questions
What does ln(x) mean in plain terms?
ln(x) is the exponent that e must be raised to in order to produce x — solve eʸ = x for y and the answer is y = ln(x). That is the natural logarithm's whole definition: the inverse of the exponential function eˣ, valid only when x is a positive real number.
Why is e the base, rather than 10?
Because e ≈ 2.718281828 is the unique number whose exponential function eˣ is its own derivative — no other base gives calculus that clean a result. Growth or decay that happens continuously, rather than in yearly or monthly steps, is described exactly by eˣ, so ln is the logarithm calculus reaches for first.
How does ln(x) relate to log base 10 or log base 2?
All logarithms are proportional to one another: ln(x) = log₁₀(x) × ln(10), and ln(10) ≈ 2.302585 is exactly the conversion constant this instrument returns for x = 10. This site's log-2 calculator performs the base-2 version of the same conversion, using ln(2) ≈ 0.693147 in place of ln(10).
Is natural log the same thing as the negative log used for pH?
No — they solve different problems. Natural log uses base e and is built for calculus and continuous growth; the negative-log used for pH and similar scales flips the sign of a base-10 logarithm to turn tiny concentrations into small positive numbers. Different bases, different purposes, not two versions of one idea.
What is the natural log of zero or a negative number?
Undefined over the real numbers. eˣ is positive for every real x, so it never reaches zero or a negative value, leaving ln nothing to invert there — ln(x) accepts only x > 0. As x shrinks toward zero from the positive side, ln(x) falls toward negative infinity rather than stopping at a finite value.
What is the most common mistake when finding a natural log by hand?
Reaching for the base-10 log key on a calculator instead of the ln key — the two differ by a factor of ln(10) ≈ 2.302585, so a mixed-up entry is off by more than double. The other frequent error is entering zero or a negative number and expecting a real result instead of the domain error it actually produces.