Natural Log of 0

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Log base
x

What is ln 0?

Answer

ln 0
does not exist

ln is the natural log, base e ≈ 2.71828.

e raised to any power is greater than zero, so no power gives 0. The closer x gets to 0, the lower ln x: ln 0.001 ≈ −6.9078.

We look for y with ey = 0.

Oxy1

The curve y = ln x passes through (1, 0) and (e, 1) and rises: for x > 1 the log is positive, for 0 < x < 1 negative.

The graph has no point at x = 0: the curve goes down along the y-axis.

Log of 0 in different bases
BaseNotationValue
eln 0does not exist
10log 0does not exist
2log₂ 0does not exist

See Also

Recalculate with different values ↑

Natural log of nearby numbers

xln x
−5does not exist
−2does not exist
−1does not exist
0does not exist
0.01≈ −4.6052
0.1≈ −2.3026
0.2≈ −1.6094

FAQ

What is ln 0?

ln 0 does not exist

What does ln x approach as x approaches 0?

It decreases without bound (to −∞): ln 0.001 ≈ −6.9078, ln 0.0001 ≈ −9.2103. There is no value at x = 0 itself.

What does log without a base mean?

At school and on a calculator, log x without a base is the common log, base 10: log 100 = 2. In Python, math.log(x) is the natural log ln x; in Excel, LN is the natural log and LOG is base 10.

See Also