Natural Logarithm Calculator
Calculate ln(x) = log base e of x with exact values, exponential form, and common values reference.
Input
Updates as you typeMode
What do you want to solve for? ?
Argument
Number x ?
—
Target
Value of ln(x) = y ?
—
Base
Target base b ?
—
y = ln(x)
Current point markedx = — · ln(x) = —
y = ln(x)
Current x
Common values
Click a row to load it| x | ln(x) | log₁₀(x) | Note |
|---|
Formula
ln(x)
=
loge(x)
,
e ≈ 2.71828…
- x
- Argument of the logarithm; must be strictly > 0
- e
- Euler's number, the unique base for which d/dx eˣ = eˣ
- ln(1) = 0
- Any logarithm of 1 is zero
- ln(e) = 1
- Logarithm of the base equals 1
Product: ln(a · b) = ln(a) + ln(b)
Quotient: ln(a / b) = ln(a) − ln(b)
Power: ln(aᵏ) = k · ln(a)
Reciprocal: ln(1/a) = −ln(a)
Derivative: d/dx ln(x) = 1/x
Integral: ∫ ln(x) dx = x · ln(x) − x + C
Series (|x − 1| ≤ 1): ln(1 + u) = u − u²/2 + u³/3 − u⁴/4 + …
Limit: limx→0⁺ ln(x) = −∞, limx→∞ ln(x) = ∞
Inverse: ln(eˣ) = x and e^(ln(x)) = x
Change of base: logb(x) = ln(x) / ln(b)
Bridge to log₁₀: log₁₀(x) = ln(x) / ln(10) ≈ ln(x) · 0.4343
Natural growth: x(t) = x₀ · e^(rt) ⇒ t = ln(x/x₀) / r
Worked example — your numbers
- Input: x = —
- ln(x) = log_e(x) where e ≈ 2.71828
- ln(—) = —
- Exponential form: —
- Bridge: log₁₀(x) = ln(x) / ln(10) ≈ —
Natural logs appear wherever quantities grow or decay continuously — radioactive decay, half-lives, continuously compounded interest, pH chemistry, and information entropy. The "natural" choice of base e makes the derivative rule clean: the slope of ln at any point x is exactly 1/x.