Natural Log Function Notes

The natural logarithm, denoted as \ln x, is the inverse function of the natural exponential function e^x.
- Inverse relationship: e^{\ln x} = x for all x>0 and \ln(e^x) = x for all real x.
- The base is the constant e\approx 2.718281828…, defined by limits such as e = \lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n or as the unique base for which the derivative of e^x equals itself.
Domain of \ln x in real numbers: x>0. Range: (-\infty, \infty).
Graphically, \ln x is increasing, passes through (1,0), has a vertical asymptote at x=0^+, and grows without bound as x\to\infty while approaching -\infty as x\to 0^+.
Practical interpretation: the natural log measures proportional (percentage) changes; it is the inverse of continuous growth modeled by e^t.

Domain: x>0; Range: (-\infty, \infty).
Key special values:
- \ln 1 = 0
- \ln e = 1
Inverse with exponential:
- e^{\ln x} = x\quad (x>0)
- \ln(e^x) = x\quad (\text{for all } x)
Fundamental rules (for a>0, b>0):
- \ln(ab) = \ln a + \ln b
- \ln\left(\frac{a}{b}\right) = \ln a - \ln b
- \ln(a^k) = k\ln a
- \ln(x^r) = r\ln x
Relationship to other bases (change of base):
- \log_b x = \dfrac{\ln x}{\ln b}
- In particular, base-10 logs relate via \log_{10} x = \dfrac{\ln x}{\ln 10}.
Domain nuance: real-valued \ln x is defined only for positive argument; expressions like \ln|x| are used in some contexts (e.g., integrals involving absolute value) but are not the natural log function itself on negative inputs.

Derivative: for x>0,
- \frac{d}{dx}\ln x = \frac{1}{x}
Integral: for x>0,
- \int \frac{1}{x}\,dx = \ln x + C
Relationship to the exponential: derivative of e^x is e^x, while the derivative of \ln x is its reciprocal; they are inverse functions.

Monotonicity: strictly increasing on (0,\infty).
Concavity: second derivative \dfrac{d^2}{dx^2}\ln x = -\dfrac{1}{x^2} < 0 for all x>0; hence concave down everywhere on its domain.
End behavior:
- As x\to 0^+, \ln x \to -\infty.
- As x\to \infty, \ln x \to \infty.
Key points:
- Passes through (1,0) and increases slowly for large x compared to polynomial functions.

Basic Taylor series around x=1 (i.e., in terms of h = x-1):
- For |h|<1,
- \ln(1+h) = h - \frac{h^2}{2} + \frac{h^3}{3} - \frac{h^4}{4} + \cdots = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{h^n}{n}.
- This gives a local approximation for values near 1, with convergence for -1 < h \le 1, i.e., 0 < x \le 2.
Common specific expansion:
- For |x|<1, \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots.
For small deviations around 1: let x = 1+t with |t|<1, then
- \ln x = t - \frac{t^2}{2} + \frac{t^3}{3} - \cdots.
Practical note: series converge more slowly as you move away from the expansion point; use change of base or other numerical methods for large values.

Quick approximate values:
- \ln 2 \approx 0.6931
- \ln 3 \approx 1.0986
- \ln 7 \approx 1.9459
- \ln 10 \approx 2.3026
Exact/inverse example:
- If \ln x = 3, then x = e^3 \approx 20.0855.

Inverse relationship with the exponential function makes \ln x central in solving equations of the form a^t = x, leading to t = \log_a x = \dfrac{\ln x}{\ln a}.
In calculus and analysis:
- Areas under the hyperbola: the area under y = \frac{1}{x} from 1 to a is \int_1^a \frac{1}{x}\,dx = \ln a.
- Growth and decay models with continuous compounding use e^{rt} and natural logs for analytic solutions.
In statistics and data science, log-transformations with base e stabilize variance and linearize multiplicative relationships.

Solve for x: \ln x = 2 ⇒ x = e^2\,.
Solve for x: \ln x = \ln 5 ⇒ x = 5\,.
Change of base: express \log3 81 using natural logs: \log3 81 = \dfrac{\ln 81}{\ln 3} = 4.
If \ln(2x) = 1, find x: 2x = e^1 = e ⇒ x = \dfrac{e}{2}.

Domain trap: never take \ln x for nonpositive x\; (x \le 0) in real-valued contexts.
Remember base-10 vs natural logarithm: many calculators have a ln button (base e) and a log button (base 10) separately; convert using \log_{10} x = \dfrac{\ln x}{\ln 10} if needed.
When solving equations involving \ln x, check that the solution lies in the domain x>0.
Use the derivative to check monotonicity: since \frac{d}{dx}\ln x = \frac{1}{x} > 0 for all x>0, the function is strictly increasing.
Be mindful of the absolute value when integrating: \int \frac{1}{x}\,dx = \ln|x| + C in a context where the domain may cross zero; for strictly positive arguments, this reduces to \ln x + C).

Basic log rules:
- \ln(ab)=\ln a+\ln b,\quad \ln\left(\frac{a}{b}\right)=\ln a-\ln b,\quad \ln(a^k)=k\ln a.
Special values:
- \ln 1 = 0,\quad \ln e = 1.
Derivative and integral:
- \frac{d}{dx}\ln x = \frac{1}{x},\quad x>0.
- \int \frac{1}{x}\,dx = \ln x + C,\quad x>0.
Change of base:
- \log_b a = \frac{\ln a}{\ln b}.
Inverse relationships:
- e^{\ln x} = x,\quad \ln(e^x) = x.
Series (around 0 for ln(1+x)):
- \ln(1+x) = \sum_{n=1}^{\infty} (-1)^{n+1}\frac{x^n}{n},\quad -1<x\le 1.$$

The natural logarithm is a fundamental, well-behaved function on its domain that links linear and exponential growth, provides a powerful tool for solving equations, computing areas, and modeling continuous processes. Mastery of its rules, derivatives, integrals, and series is essential for core calculus, analysis, and applied contexts.