Calculus: Analytic Geometry and Calculus, with Vectors

486 Exponential and logarithmic functions For more than 300 years, logarithms with base 10 were systematically and extensively u ...

9.1 Exponentials and logarithms 487 the graph of y = loge x at the point (1,0). Remark: This is not a dull problem, because mode ...

488 Exponential and logarithmic functions Remark: The formulas (1) and (3) and (4) are examples of Stirling formulas for log at ...

9.1 Exponentials and logarithms 489 17 Persons who start fires with matchescan be interested in the fact that fires can be start ...

490 Exponential and logarithmic functions The LL3 scale contains numbers from e to about 22,000. The distance from the left end ...

9.2 Derivatives and integrals of exponentialsand logarithms 491 But (H''Q)XP = (H),P)xQ and (HxQ)P = (H>P)Q, and it follows t ...

492 Exponential and logarithmic functions To investigate the factor ah - 1, we employ the method which we used to prove Theorem ...

9.2 Derivatives and integrals of exponentials and logarithms 493 Since a0 has a positive derivative, Theorem 8.33 implies that i ...

494 Y 1 01 1 z Figure 9.263 e3 4 X correct to 5 or 10 decimal places Exponential and logarithmic functions which we can now obta ...

9.2 Derivatives and integrals of exponentialsand logarithms 495 Theorem 9.271 If p is a constant positiveexponent, then the firs ...

(^496) Exponential and logarithmic functions is a continuous increasing function for which L(e) = 1, the formula (9.281) must be ...

9.2 Derivatives and integrals of exponentials and logarithms 497 t (time) in such a way that its rate of change with respect tot ...

498 Exponential and logarithmic functions With or without more attention to details, jump to the conclusion that, for each n=1, ...

9.2 Derivatives and integrals of exponentials and logarithms^499 Note that considerations very similar to those in Problem 8 est ...

500 Hence prove that (7) Exponential and logarithmic functions log11x=x+2+3+3+... when -1 S x < 1 and that (8) log2=1+ +3-1 . ...

9.2 Derivatives and integrals of exponentials and logarithms 501 tive, we can begin by being irked by the fact that the base is ...

(^502) Exponential and logarithmic functions (^18) Derive the formula fez +1dx=x-log(ex+1)+c with the aid of the identity I ez+1 ...

9.2 Derivatives and integrals of exponentials and logarithms 503 F and each of the derivatives F(l), n = 1, 2,3, ,is continuous ...

504 Exponential and logarithmic functions Since fi(al) = 1, this gives the remarkable parade` of inequalities (8) 1 G Cal + Qn\ ...

9.3 Hyperbolic functions 505 by putting s, = f(n) in Theorem 5.651, this and (5) imply existence of a number L for which (6) lim ...