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 = loga x 的導數，得 dy/dx = (1/x) loga e ，當 a=e 時， loge 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。

若將指數函數 ex 作泰勒展開，則得

ex = 1 + x + x2 + x3 + x4 + … 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

The exponential function f with base a is denoted by f(x)=ax, 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.

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

Exponential functions Definition

Take a > 0 and not equal to 1 . Then, the function defined by f : R -> R : x -> ax

is called an exponential function with base a.

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

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

Logarithmic functions

Definition and basic properties

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

loga(x) = y <=> ay = x

for x > 0 we have aloga(x) = x for all x we have loga(ax) = x

GraphLet f(x) = a logarithmic function with a > 1.Let g(x) = a logarithmic function with 0 < a < 1.

log(x.y) = log(x) + log(y)

log(x/y) = log(x) - log(y)

log(xr ) = r.log(x)

Pf:

log(x.y) = u then au = x.y (1)

Let log(x) = v then av = x (2)

Let log(y) = w then aw = y (3)

From (1) , (2) and (3)

au = av . aw => au = av + w => u = v + w

Change the base of a logarithmic fun.Theorem:for each strictly positive real number a and b, different from 1,

loga(x) =( ) .

logb(x)logb(a)

1

loga(xr) = r logax

log22 = 1

log31 = 0

log25 = log 5    

log 2

  2.3219

Identity Example

loga(xy) = logax + logay log216 = log28 +

log22loga(x/y) = logax - logay log2 (5/3) = log25 -

log23

log2(65) = 5 log26

logaa = 1

loga1 = 0

loga(1/x) = -logax log2(1/3)= -log23

logax = log x    

log a

= ln x   ln a

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

alogax= x

for all positive x and loga(ax) = x for all real x.

Ex

2log2x=

eln x=

log2(2x) =

ln (ex) =

Ans: x

Definition of Logarithmic Function

For x >0, a>0 , and a ≠ 1, we have f(x)=loga(x) iff a f(x) =x Since x > 0, the graph of the

above function will be in quadrants I and IV.

Comments on Logarithmic Functions

The exponential equation 43=64, could be written in terms of a logarithmic equation as log4(64)=3.

The exponential equation 5-2=1/25 can be written as the logarithmic equation log5(1/25)=-2.

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.

The derivatives of the logarithmic functions

Derivative of logb and ln

An important special case is this:

d/dx logb(x) = 1 / x ln (b)

d/dx ln (x) = 1/x since ln e =1

Derivative of bx and ex

(d/dx) bx = bx ln(b)

Ex: d/dx [e ] 4x2-2

Ex:

d ex - e-x

dx ex + e-x

Ex: d/dx 2x(4 x ) = 2(4 x ) +2x(4 x ) ln4

Ex: d/dx ln (x2 + 2x -1)

Ex: d/dx ln (3x + 2)

Ex: d/dx log 3 (x) = 1 / x ln (3)

(3x + 2)