Natural Log Calculator — ln(x)
Calculate the natural logarithm ln(x) (base e), log₁₀(x), log₂(x), the inverse eˣ, and the slope 1/x at your chosen point — with the ln curve plotted.
Natural logarithm — the power to which e ≈ 2.71828 must be raised to equal x
- 1
log₁₀(x)
log₁₀(1) = 0 - 2
Convert to natural log
0 × ln(10) = 0Change of base: ln(x) = log₁₀(x) × ln(10) ≈ log₁₀(x) × 2.302585.
How does this calculator work?
ln(x) is the natural logarithm with base e ≈ 2.71828. It is the inverse of eˣ, is only defined for x > 0, equals 0 at x = 1, and its derivative is 1/x. For other bases: log₁₀(x) = ln(x)/ln(10), log₂(x) = ln(x)/ln(2). The curve rises steeply near zero and flattens for large x.
Formula
How this is calculated
The natural logarithm ln(x) is the logarithm with base e ≈ 2.71828182845, the mathematical constant at the heart of continuous growth, compound interest, and probability theory. It answers: "to what power must e be raised to equal x?" — so ln(e) = 1, ln(1) = 0, and ln(x) is negative for 0 < x < 1. The function is only defined for strictly positive real x.
All logarithms in any base are proportional via the change-of-base rule: log_b(x) = ln(x) / ln(b). This calculator applies it to give log₁₀(x) = ln(x)/ln(10) (useful in chemistry and dB calculations) and log₂(x) = ln(x)/ln(2) (used in information theory and computer science). The inverse of ln is the natural exponential: e^ln(x) = x and ln(eˣ) = x.
The derivative of ln(x) is 1/x — the slope of the curve at any point x equals the reciprocal of x. This means the curve rises steeply near zero and almost flattens for large x, captured in the XY plot. The marked point shows exactly where your chosen x sits on this curve.
Frequently asked questions
ln always means the natural logarithm (base e ≈ 2.718). "log" is ambiguous: in engineering and many calculators it means log base 10; in mathematics textbooks it often means ln; in computing it can mean log base 2. This calculator shows all three explicitly.
Any number raised to the power 0 equals 1, so e⁰ = 1 by definition, which means ln(1) = 0. Similarly, ln(e) = 1 because e¹ = e. For x < 1 (but x > 0), ln(x) is negative because e must be raised to a negative power to produce a value less than 1.
Not in the real numbers — ln(x) is undefined for x ≤ 0. In complex analysis, ln(−1) = iπ (Euler's famous identity), but for practical real-number work the input must be strictly positive. Entering x ≤ 0 in this calculator returns a warning rather than a result.
Also known as
TG we-Calculate Editorial Team. (2026). Natural Log Calculator — ln(x) [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/natural-log-calculator
TG we-Calculate Editorial Team. "Natural Log Calculator — ln(x)." TG we-Calculate. 2026. https://we-calculate.com/calculator/natural-log-calculator.
TG we-Calculate Editorial Team, "Natural Log Calculator — ln(x)," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/natural-log-calculator
@misc{wecalculate_natural_log_calculator, title = {Natural Log Calculator — ln(x)}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/natural-log-calculator}}, year = {2026}, note = {TG we-Calculate} }
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