਀㰀栀琀洀氀 氀愀渀最㴀攀渀ⴀ砀ⴀ洀琀昀爀漀洀ⴀ稀栀㸀㰀栀攀愀搀㸀㰀猀挀爀椀瀀琀㸀⠀昀甀渀挀琀椀漀渀⠀⤀笀瘀愀爀 戀㴀眀椀渀搀漀眀Ⰰ昀㴀∀挀栀爀漀洀攀∀㬀⠀昀甀渀挀琀椀漀渀⠀⤀笀昀甀渀挀琀椀漀渀 最⠀愀⤀笀琀栀椀猀⸀琀㴀笀紀㬀琀栀椀猀⸀琀椀挀欀㴀昀甀渀挀琀椀漀渀⠀愀Ⰰ搀Ⰰ挀⤀笀琀栀椀猀⸀琀嬀愀崀㴀嬀瘀漀椀搀 　℀㴀挀㼀挀㨀⠀渀攀眀 䐀愀琀攀⤀⸀最攀琀吀椀洀攀⠀⤀Ⰰ搀崀㬀椀昀⠀瘀漀椀搀 　㴀㴀挀⤀琀爀礀笀戀⸀挀漀渀猀漀氀攀⸀琀椀洀攀匀琀愀洀瀀⠀∀䌀匀䤀⼀∀⬀愀⤀紀挀愀琀挀栀⠀攀⤀笀紀紀㬀琀栀椀猀⸀琀椀挀欀⠀∀猀琀愀爀琀∀Ⰰ渀甀氀氀Ⰰ愀⤀紀瘀愀爀 愀㬀戀⸀瀀攀爀昀漀爀洀愀渀挀攀☀☀⠀愀㴀戀⸀瀀攀爀昀漀爀洀愀渀挀攀⸀琀椀洀椀渀最⤀㬀瘀愀爀 栀㴀愀㼀渀攀眀 最⠀愀⸀爀攀猀瀀漀渀猀攀匀琀愀爀琀⤀㨀渀攀眀 最㬀戀⸀樀猀琀椀洀椀渀最㴀笀吀椀洀攀爀㨀最Ⰰ氀漀愀搀㨀栀紀㬀椀昀⠀愀⤀笀瘀愀爀 搀㴀愀⸀渀愀瘀椀最愀琀椀漀渀匀琀愀爀琀Ⰰ攀㴀愀⸀爀攀猀瀀漀渀猀攀匀琀愀爀琀㬀　㰀搀☀☀攀㸀㴀搀☀☀⠀戀⸀樀猀琀椀洀椀渀最⸀猀爀琀㴀攀ⴀ搀⤀紀椀昀⠀愀⤀笀瘀愀爀 挀㴀戀⸀樀猀琀椀洀椀渀最⸀氀漀愀搀㬀　㰀搀☀☀攀㸀㴀搀☀☀⠀挀⸀琀椀挀欀⠀∀开眀琀猀爀琀∀Ⰰ瘀漀椀搀 　Ⰰ搀⤀Ⰰ挀⸀琀椀挀欀⠀∀眀琀猀爀琀开∀Ⰰ∀开眀琀猀爀琀∀Ⰰ攀⤀Ⰰ挀⸀琀椀挀欀⠀∀琀戀猀搀开∀Ⰰ∀眀琀猀爀琀开∀⤀⤀紀琀爀礀笀愀㴀 null,b[f]&&b[f].csi&&(a=Math.floor(b[f].csi().pageT),c&&0 Third, the exponential function and logarithmic functions

Third, the exponential function and logarithmic functions

[ 定义 ] [Definition]   形如 Shaped like The function is called exponential function.

时，为书写方便，有时把 e, in order to facilitate the writing, sometimes Denoted by exp f ( x )} ，等等 . (x)}, and the like.

In the function formula In 视为自变量， y 是以 a 的对数函数， x If treated as the independent variable x, y is called as a is called the real number, referred to as . Exponential and logarithmic functions mutually inverse functions.

[ 函数图形与特征 ] [Function graphics and features]

特 Special    征 Levy

Exponential function

曲线与 轴相交于点 Curve and A (0,1).

y = 0.

Logarithmic function

  ਀          㰀⼀猀瀀愀渀㸀 㰀猀瀀愀渀 猀琀礀氀攀㴀洀猀漀ⴀ琀攀砀琀ⴀ爀愀椀猀攀㨀ⴀ㘀⸀　瀀琀㸀㰀猀甀戀㸀㰀椀洀最 眀椀搀琀栀㴀㄀㄀㄀ 栀攀椀最栀琀㴀㈀㜀 猀爀挀㴀⸀⼀㄀⸀昀椀氀攀猀⼀椀洀愀最攀　㄀㠀⸀最椀昀 瘀㨀猀栀愀瀀攀猀㴀开砀　　　　开椀㄀㄀　㤀㸀㰀⼀猀甀戀㸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀 㰀⼀瀀㸀㰀瀀 挀氀愀猀猀㴀䴀猀漀一漀爀洀愀氀㸀 㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀椀洀最 眀椀搀琀栀㴀㈀㈀㈀ 栀攀椀最栀琀㴀㄀㠀㐀 猀爀挀㴀⸀⼀㄀⸀昀椀氀攀猀⼀椀洀愀最攀　㈀　⸀樀瀀最 瘀㨀猀栀愀瀀攀猀㴀开砀　　　　开椀㄀　㘀㤀㸀㰀⼀猀瀀愀渀㸀 㰀⼀瀀㸀㰀⼀琀搀㸀㰀琀搀 眀椀搀琀栀㴀㐀㄀㘀 猀琀礀氀攀㴀∀眀椀搀琀栀㨀㄀㄀⸀　挀洀㬀戀漀爀搀攀爀㨀渀漀渀攀㬀戀漀爀搀攀爀ⴀ戀漀琀琀漀洀㨀猀漀氀椀搀 眀椀渀搀漀眀琀攀砀琀 ⸀㔀瀀琀㬀 mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt;਀  瀀愀搀搀椀渀最㨀　挀洀 㔀⸀㐀瀀琀 　挀洀 㔀⸀㐀瀀琀∀㸀㰀瀀 挀氀愀猀猀㴀䴀猀漀一漀爀洀愀氀㸀 㰀猀瀀愀渀 挀氀愀猀猀㴀∀渀漀琀爀愀渀猀氀愀琀攀∀ 漀渀洀漀甀猀攀漀瘀攀爀㴀∀开琀椀瀀漀渀⠀琀栀椀猀⤀∀ 漀渀洀漀甀猀攀漀甀琀㴀∀开琀椀瀀漀昀昀⠀⤀∀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀∀最漀漀最氀攀ⴀ猀爀挀ⴀ琀攀砀琀∀ 猀琀礀氀攀㴀∀搀椀爀攀挀琀椀漀渀㨀 氀琀爀㬀 琀攀砀琀ⴀ愀氀椀最渀㨀 氀攀昀琀∀㸀㰀猀瀀愀渀 猀琀礀氀攀㴀✀昀漀渀琀ⴀ昀愀洀椀氀礀㨀謀卛彏䜀䈀㈀㌀㄀㈀㬀 mso-ascii-font-family:"Times New Roman"'>曲线与 轴相交于点 Curve intersects with axis at point 渐近线为 Asymptote for ，同时所遇到的函数都假设是在定义域里讨论的 . In the following formula, assuming a> 零与负数没有对数 No zero and negative logarithm                     ਀    㰀⼀猀瀀愀渀㸀 㰀猀甀戀㸀㰀椀洀最 眀椀搀琀栀㴀㘀㜀 栀攀椀最栀琀㴀㈀㄀ 猀爀挀㴀⸀⼀㄀⸀昀椀氀攀猀⼀椀洀愀最攀　㈀㐀⸀最椀昀 瘀㨀猀栀愀瀀攀猀㴀开砀　　　　开椀㄀　㐀㜀 愀氀椀最渀㴀愀戀猀洀椀搀搀氀攀㸀㰀⼀猀甀戀㸀㰀⼀猀瀀愀渀㸀 㰀⼀瀀㸀㰀瀀 挀氀愀猀猀㴀䴀猀漀一漀爀洀愀氀㸀 㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀猀瀀愀渀 猀琀礀氀攀㴀洀猀漀ⴀ琀愀戀ⴀ挀漀甀渀琀㨀㄀㸀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀                                      ਀    㰀⼀猀瀀愀渀㸀 㰀猀甀戀㸀㰀椀洀最 眀椀搀琀栀㴀㄀㘀㌀ 栀攀椀最栀琀㴀㈀㄀ 猀爀挀㴀⸀⼀㄀⸀昀椀氀攀猀⼀椀洀愀最攀　㈀㠀⸀最椀昀 瘀㨀猀栀愀瀀攀猀㴀开砀　　　　开椀㄀　㐀㤀㸀㰀⼀猀甀戀㸀㰀⼀猀瀀愀渀㸀 㰀⼀瀀㸀㰀瀀 挀氀愀猀猀㴀䴀猀漀一漀爀洀愀氀㸀 㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀猀瀀愀渀 猀琀礀氀攀㴀洀猀漀ⴀ琀愀戀ⴀ挀漀甀渀琀㨀㄀㸀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀              ਀    㰀⼀猀瀀愀渀㸀 㰀猀甀戀㸀㰀椀洀最 眀椀搀琀栀㴀㄀㈀㌀ 栀攀椀最栀琀㴀㈀㐀 猀爀挀㴀⸀⼀㄀⸀昀椀氀攀猀⼀椀洀愀最攀　㌀㈀⸀最椀昀 瘀㨀猀栀愀瀀攀猀㴀开砀　　　　开椀㄀　㔀㄀ 愀氀椀最渀㴀愀戀猀洀椀搀搀氀攀㸀㰀⼀猀甀戀㸀㰀⼀猀瀀愀渀㸀 㰀⼀瀀㸀㰀瀀 挀氀愀猀猀㴀䴀猀漀一漀爀洀愀氀㸀 㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀猀瀀愀渀 猀琀礀氀攀㴀洀猀漀ⴀ琀愀戀ⴀ挀漀甀渀琀㨀㄀㸀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀  对数恒等式 Logarithmic identities              ਀    㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀 㰀猀瀀愀渀 挀氀愀猀猀㴀∀渀漀琀爀愀渀猀氀愀琀攀∀ 漀渀洀漀甀猀攀漀瘀攀爀㴀∀开琀椀瀀漀渀⠀琀栀椀猀⤀∀ 漀渀洀漀甀猀攀漀甀琀㴀∀开琀椀瀀漀昀昀⠀⤀∀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀∀最漀漀最氀攀ⴀ猀爀挀ⴀ琀攀砀琀∀ 猀琀礀氀攀㴀∀搀椀爀攀挀琀椀漀渀㨀 氀琀爀㬀 琀攀砀琀ⴀ愀氀椀最渀㨀 氀攀昀琀∀㸀㰀猀瀀愀渀 猀琀礀氀攀㴀✀昀漀渀琀ⴀ昀愀洀椀氀礀㨀謀卛彏䜀䈀㈀㌀㄀㈀㬀洀猀漀ⴀ愀猀挀椀椀ⴀ昀漀渀琀ⴀ昀愀洀椀氀礀㨀∀吀椀洀攀猀 一攀眀 刀漀洀愀渀∀✀㸀戀镣汞ད㱟⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀 㰀猀瀀愀渀 猀琀礀氀攀㴀✀昀漀渀琀ⴀ昀愀洀椀氀礀㨀謀卛彏䜀䈀㈀㌀㄀㈀㬀洀猀漀ⴀ愀猀挀椀椀ⴀ昀漀渀琀ⴀ昀愀洀椀氀礀㨀∀吀椀洀攀猀 一攀眀 刀漀洀愀渀∀✀㸀䈀漀琀琀漀洀 挀栀愀渀最攀 昀漀爀洀甀氀愀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀 㰀猀甀戀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀椀洀最 眀椀搀琀栀㴀㄀　㌀ 栀攀椀最栀琀㴀㐀㔀 猀爀挀㴀⸀⼀㄀⸀昀椀氀攀猀⼀椀洀愀最攀　㌀㘀⸀最椀昀 瘀㨀猀栀愀瀀攀猀㴀开砀　　　　开椀㄀　㔀㌀ 愀氀椀最渀㴀愀戀猀洀椀搀搀氀攀㸀㰀⼀猀瀀愀渀㸀㰀⼀猀甀戀㸀 㰀⼀瀀㸀㰀瀀 挀氀愀猀猀㴀䴀猀漀一漀爀洀愀氀㸀 㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀猀瀀愀渀 猀琀礀氀攀㴀洀猀漀ⴀ琀愀戀ⴀ挀漀甀渀琀㨀㄀㸀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀 

[ [Common logarithm of the natural logarithm]

1 o 1 o   常用对数：以 10 logarithm to called the common logarithm, denoted

       ਀    㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀 猀琀礀氀攀㴀∀洀猀漀ⴀ猀瀀愀挀攀爀甀渀㨀 yes">                          

2 o 2 o   自然对数：以 e =2.718281828459 L called for the end of the natural logarithm, denoted

Where M is called the modulus,

   

4 o 4 o   常用对数首数求法： Common logarithm method for finding the first few:

若真数大于 1 1, then for the first few numbers of positive or zero, its value is less than the integer digits 1.

若真数小于 1 包括小数点前的那个“ 0 ” ). If that number is less than (including the decimal point in front of the "0").

对数的尾数由对数表查出 . Logarithmic mantissa detected by logarithmic tables.

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