Natural Logarithm ln(x): Graph, Rules, Derivative and Integral

The natural logarithm \ln x answers the question "what power of e gives x?". It is the inverse of the natural exponential function, and its derivative is the remarkably simple \frac{1}{x}, which makes it the logarithm of choice in calculus.

Definition and formula

Equivalently, \ln x is the area under the reciprocal function from 1 to x:

Graph of ln x

Key properties

• Domain (0, \infty), range all reals.
• Key values: \ln 1 = 0, \ln e = 1, \ln e^k = k.
• Increasing everywhere (\frac{1}{x} > 0) and concave down (-\frac{1}{x^2} < 0).
• Vertical asymptote x = 0: \lim_{x \to 0^+}\ln x = -\infty. Also \lim_{x \to \infty}\ln x = \infty, but very slowly; \ln x grows slower than any positive power of x.
• Inverse identities: e^{\ln x} = x for x > 0 and \ln e^x = x for all x.

Log rules

These hold for a, b > 0. There is no rule for \ln(a + b).

Derivative of ln x

Proof sketch: if y = \ln x then e^y = x. Differentiating implicitly gives e^y y' = 1, so y' = \frac{1}{e^y} = \frac{1}{x}. With the chain rule:

Integral of ln x

Using integration by parts with u = \ln x, dv = dx, so du = \frac{1}{x}\,dx and v = x:

The reverse direction is the more common one: \int \frac{1}{x}\,dx = \ln|x| + C.

Worked examples

Example 1: chain rule. \frac{d}{dx}\ln(x^2 + 1) = \frac{2x}{x^2 + 1}.

Example 2: simplify first. \frac{d}{dx}\ln\frac{x^3}{x + 1} = \frac{d}{dx}\left[3\ln x - \ln(x + 1)\right] = \frac{3}{x} - \frac{1}{x + 1}.

Example 3: product rule. \frac{d}{dx}(x^2\ln x) = 2x\ln x + x.

Example 4: definite integral. \int_1^e \ln x\,dx = \left[x\ln x - x\right]_1^e = (e - e) - (0 - 1) = 1.

Example 5: logarithmic integral. With u = x^2 + 4, \int \frac{2x}{x^2 + 4}\,dx = \ln(x^2 + 4) + C. No absolute value is needed since x^2 + 4 > 0.

Common mistakes

1. Splitting a log of a sum. \ln(x + 1) \ne \ln x + \ln 1.
2. Writing \frac{d}{dx}\ln(5x) = \frac{1}{5x}. The chain rule gives \frac{5}{5x} = \frac{1}{x}.
3. Writing \int \ln x\,dx = \frac{1}{x} + C. That is the derivative, not the integral.
4. Confusing \ln x^2 with (\ln x)^2. The first is 2\ln x; the second is a square.
5. Taking \ln of 0 or a negative number. Check the domain, especially after a substitution.

How it connects

Taking \ln of both sides turns hard products and powers into sums, the idea behind logarithmic differentiation. Logs with other bases are just rescaled natural logs; see logarithmic functions. For more worked practice, read the guide to derivatives of exponential and logarithmic functions.