Mathematics · Logarithms

What is a natural logarithm?

ln(x) gives the step number on a continuous scale where each whole step multiplies by e.

One full step multiplies by e.

Starting at 1, one full step gives e ≈ 2.718, and two full steps give e2 ≈ 7.389.

Whole steps on an e-based scale Step zero has value one. Step one has value e, about 2.718. Step two has value e squared, about 7.389. Each arrow multiplies by e. × e× e 12.7187.389 step 0step 1step 2
Illustrative scale · displayed decimals rounded
What a “step” means
  • A step number s is a real exponent: x = es.
  • Euler’s number e begins 2.718281828….

ln reads the step number.

ln(x) returns the exponent s in es = x.

Natural logarithm playground The horizontal axis is the positive value x. The vertical axis is the step number ln(x). The marker starts at x equals one and ln(x) equals zero.
Illustrative graph · x = es
x (2 decimals)1.00
ln(x) (3 decimals)0.000

Before moving below 1, predict the sign of ln(x).

Slider range and rounding
  • The slider covers 0.20 to 8.00 in increments of 0.01.
  • The point uses ln(x) before rounding; the readouts show two and three decimal places.
  • The slider may not land exactly on e.

Half a step uses √e.

Two half-steps multiply by √e twice, giving one full factor of e.

Two half-steps make one full step Start at one. Half a step gives square root of e, about 1.649. Another half-step gives e, about 2.718. × √e× √e 11.6492.718 step 0step 0.5step 1
Illustrative scale · ln(√e) = 0.5
Why half a step is a square root
  • e0.5 = √e.
  • √e × √e = e, so two moves of 0.5 give a move of 1.
  • A fractional step is a real exponent, not a count of whole multiplications.

Going back one step divides by e.

A positive value below 1 has a negative step number because e raised to a negative exponent is below 1.

Negative steps give positive values below one From right to left: one at step zero, one divided by e at step minus one, and one divided by e squared at step minus two. All three values are positive. ÷ e÷ e 0.1350.3681 step −2step −1step 0
Positive x only · ln(0) and ln(negative x) have no real value
Negative steps and the input limit
  • e−1 = 1/e ≈ 0.368, so ln(1/e) = −1.
  • es stays positive for every real step number s.
  • The real natural logarithm therefore accepts only x > 0.

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