2. e 的發現始於微分，當 h 逐漸接近零時，計 算 (1+h) 1/h 之值，其結果無限接近一定值 2.71828... ，這個定值就是 e ，最早發現此值 的人是瑞士著名數學家歐拉，他以自己姓名 的字頭小寫 e 來命名此無理數。 計算對數函數 y = log a x 的導數，得 dy/dx = (1/x) log a e ，當 a=e 時， log e x 的導 數為 1/x ，因而有理由使用以 e 為底的對數，這 叫作自然對數。

3. 若將指數函數 e x 作泰勒展開，則得 e x = 1 + x + x 2 + x 3 + x 4 + … 2! 3! 4! 以 x=1 代入上式得 e = 1 + 1 + ½ + 1/6 + 1/12 +…. 此級數收斂迅速， e 近似到小數點後 40 位 的數值是 2.71828 18284 59045 23536 02874

4. The exponential function f with base a is denoted by f(x)=a x, where a≠1, and x is any real number. The function value will be positive because a positive base raised to any power is positive.

5. Ex: if the base is 2 and x = 4, the function value f(4) will equal 16. The graph of f(x)=2 x would be (4, 16).

6. Exponential functions Definition Take a > 0 and not equal to 1. Then, the function defined by f : R -> R : x -> a x is called an exponential function with base a.

7. Graph and properties Let f(x) = an exp. fun. with a > 1. Let g(x) = an exp. Fun. with 0 < a < 1.

8. From the graphs we see that  The domain is R  The range is the set of strictly  positive real numbers  The function is continuous in its domain  The function is increasing if a > 1 and decreasing if 0 < a < 1  The x-axis is a horizontal asymptote

9. Logarithmic functions Definition and basic properties Take a > 0 and not equal to 1. Since the exponential function f : R -> R : x -> a x are either increasing or decreasing, the inverse function is defined. This inverse function is called the logarithmic function with base a. We write log a (x)

10. log a (x) = y a y = x for x > 0 we have alog a (x) = x for all x we have log a (ax) = x Graph Let f(x) = a logarithmic function with a > 1. Let g(x) = a logarithmic function with 0 < a < 1.

12. log(x.y) = log(x) + log(y) log(x/y) = log(x) - log(y) log(xr ) = r.log(x)

13. Pf: log(x.y) = u then a u = x.y (1) Let log(x) = v then a v = x (2) Let log(y) = w then a w = y (3) From (1), (2) and (3) a u = a v. a w => a u = a v + w => u = v + w

14. Change the base of a logarithmic fun. Theorem:for each strictly positive real number a and b, different from 1, log a (x) =( ). log b (x) log b (a) 1

15. log a (x r )=r log a x log 2 2=1 log 3 1=0 log 2 5 =log 5 log 2 2.3219 Identity Example log a (xy)=log a x + log a ylog 2 16 = log 2 8 + log 2 2 log a (x/y)=log a x - log a ylog 2 (5/3) = log 2 5 - log 2 3 log 2 (6 5 ) = 5 log 2 6 log a a=1 log a 1=0 log a (1/x)=-log a xlog 2 (1/3)= -log 2 3 log a x =log x log a =ln x ln a

16. Relationship of the Functions f(x) = log a x and g(x) = a x If a is any positive number, then the functions f(x) = log a x and g(x) = a x are inverse functions. This means that a logax = x for all positive x and log a (a x ) = x for all real x.

17. Ex 2 log2x = e ln x = log 2 (2 x ) = ln (e x ) = Ans: x

18. Definition of Logarithmic Function For x >0, a>0, and a ≠ 1, we have f(x)=log a (x) iff a f(x) =x Since x > 0, the graph of the above function will be in quadrants I and IV.

20. Comments on Logarithmic Functions The exponential equation 4 3 =64, could be written in terms of a logarithmic equation as log 4 (64)=3. The exponential equation 5 -2 =1/25 can be written as the logarithmic equation log 5 (1/25)=-2.

21. Logarithmic functions are the inverse of exponential functions. For example if (4, 16) is a point on the graph of an exponential function, then (16, 4) would be the corresponding point on the graph of the inverse logarithmic function.

23. The derivatives of the logarithmic functions Derivative of log b and ln An important special case is this: d/dx log b (x) = 1 / x ln (b) d/dx ln (x) = 1/x since ln e =1 Derivative of b x and e x (d/dx) b x = b x ln(b)

24. Ex: d/dx [ e ] 4x 2 -2 Ex: d e x - e -x dx e x + e -x Ex: d/dx 2x(4 x ) = 2(4 x ) +2x(4 x ) ln4

25. Ex: d/dx ln (x 2 + 2x -1) Ex: d/dx ln (3x + 2) Ex: d/dx log 3 (x) = 1 / x ln (3) (3x + 2)