log x is at most x - 1

Reading path

To understand this page from the ground up, read these 66 pages in order. The foundations come first.

1. Convex function definition
2. Limit point of a set definition
3. Local maximum definition
4. Sequence definition
5. Strictly monotone function definition
6. Slopes of a convex function are increasing statement
7. Limit of a function definition
8. Local minimum definition
9. Real sequence definition
10. Subsequence definition
11. Continuity at a point definition
12. Derivative definition
13. One-sided limit definition
14. Bounded sequence definition
15. Geometric sequence definition
16. Limit of a sequence definition
17. Monotone sequence definition
18. Series definition
19. Continuous function definition
20. Differentiable function definition
21. Fermat's theorem on stationary points statement
22. Sum of a finite geometric series statement
23. Monotone convergence theorem for sequences statement
24. Convergent sequence definition
25. Every sequence has a monotone subsequence statement
26. Squeeze theorem statement
27. Cauchy product definition
28. Partial sum definition
29. Power series definition
30. Intermediate value theorem statement
31. Derivative function definition
32. One-sided limits of a monotone function exist statement
33. Bolzano–Weierstrass theorem statement
34. The sequence r^n tends to zero for |r| < 1 example
35. Radius of convergence definition
36. Convergent series definition
37. Higher derivative definition
38. Boundedness theorem statement
39. Continuity of the inverse of a strictly monotone continuous function statement
40. A series of nonnegative terms converges iff its partial sums are bounded statement
41. Absolute convergence definition
42. Sum of a series definition
43. Second derivative definition
44. Extreme value theorem statement
45. Comparison test statement
46. Geometric series example
47. Cauchy product of absolutely convergent series statement
48. Rolle's theorem statement
49. Ratio test statement
50. Mean value theorem statement
51. Ratio formula for the radius of convergence statement
52. A differentiable function is convex iff its derivative is increasing statement
53. A function with nonnegative derivative is increasing statement
54. A function with positive derivative is strictly increasing statement
55. The exponential series has infinite radius of convergence statement
56. A function with nonnegative second derivative is convex statement
57. Exponential function definition
58. Exponential of a sum statement
59. The exponential function is positive statement
60. The exponential function is its own derivative statement
61. The exponential function is convex statement
62. The exponential function is strictly increasing statement
63. 1 + x is at most exp x statement
64. Natural logarithm definition
65. Logarithm is the inverse of the exponential statement
66. log x is at most x - 1 statement