对数法则

以 b 为底数的对数是一个指数，表示我们需要将底数 b 提升多少次方才能得到该数。

对数定义

当 b 的 y 次方等于 x 时:

b y = x

那么以 b 为底 x 的对数等于 y:

logb(x) = y

例如:

24 = 16

那么

log2(16) = 4

对数与指数函数互为反函数

对数函数,

y = logb(x)

是指数函数的反函数,

x = by

So if we calculate the exponential function of the logarithm of x (x>0),

f (f -1(x)) = blogb(x) = x

Or if we calculate the logarithm of the exponential function of x,

f -1(f (x)) = logb(bx) = x

自然对数 (ln)

自然对数是以e为底的对数:

ln(x) = loge(x)

When e constant is the number:

e=\lim_{x\rightarrow \infty }\left ( 1+\frac{1}{x} \right )^x = 2.718281828459...

or

e=\lim_{x\rightarrow 0 }\left ( 1+ \right x)^\frac{1}{x}


See: 自然对数

Inverse logarithm calculation

The inverse logarithm (or anti logarithm) is calculated by raising the base b to the logarithm y:

x = log-1(y) = b y

对数函数

对数函数的基本形式为:

f (x) = logb(x)

对数运算法则

Rule name Rule
对数乘积法则
 logb(x ∙ y) = logb(x) + logb(y)
对数除法法则
 logb(x / y) = logb(x) - logb(y)
对数幂法则
 logb(x y) = y ∙ logb(x)
对数换底法则
 logb(c) = 1 / logc(b)
对数换底公式
 logb(x) = logc(x) / logc(b)
对数导数
 f (x) = logb(x) ⇒ f ' (x) = 1 / ( x ln(b) )
对数积分
 ∫ logb(x) dx = x ∙ ( logb(x) - 1 / ln(b) ) + C
负数的对数
 当 x≤ 0 时，logb(x) 未定义
0的对数
 logb(0) 未定义
\lim_{x\to 0^+}\textup{log}_b(x)=-\infty
1的对数
 logb(1) = 0
底数的对数
 logb(b) = 1
无穷大对数
 lim logb(x) = ∞, 当 x→∞

查看: 对数规则


对数乘积法则

The logarithm of the multiplication of x and y is the sum of logarithm of x and logarithm of y.

logb(x ∙ y) = logb(x) + logb(y)

For example:

log10(3 ∙ 7) = log10(3) + log10(7)

对数除法法则

The logarithm of the division of x and y is the difference of logarithm of x and logarithm of y.

logb(x / y) = logb(x) - logb(y)

For example:

log10(3 / 7) = log10(3) - log10(7)

对数幂法则

The logarithm of x raised to the power of y is y times the logarithm of x.

logb(x y) = y ∙ logb(x)

For example:

log10(28) = 8∙ log10(2)

对数换底法则

The base b logarithm of c is 1 divided by the base c logarithm of b.

logb(c) = 1 / logc(b)

For example:

log2(8) = 1 / log8(2)

对数换底公式

The base b logarithm of x is base c logarithm of x divided by the base c logarithm of b.

logb(x) = logc(x) / logc(b)

For example, in order to calculate log2(8) in calculator, we need to change the base to 10:

log2(8) = log10(8) / log10(2)

See: log base change rule

Logarithm of negative number

The base b real logarithm of x when x<=0 is undefined when x is negative or equal to zero:

logb(x) is undefined when x ≤ 0

See: log of negative number

Logarithm of 0

The base b logarithm of zero is undefined:

logb(0) is undefined

The limit of the base b logarithm of x, when x approaches zero, is minus infinity:

\lim_{x\to 0^+}\textup{log}_b(x)=-\infty

See: log of zero

Logarithm of 1

The base b logarithm of one is zero:

logb(1) = 0

For example, teh base two logarithm of one is zero:

log2(1) = 0

See: log of one

Logarithm of infinity

The limit of the base b logarithm of x, when x approaches infinity, is equal to infinity:

lim logb(x) = ∞, when x→∞

See: log of infinity

Logarithm of the base

The base b logarithm of b is one:

logb(b) = 1

For example, the base two logarithm of two is one:

log2(2) = 1

Logarithm derivative

When

f (x) = logb(x)

Then the derivative of f(x):

f ' (x) = 1 / ( x ln(b) )

See: log derivative

Logarithm integral

The integral of logarithm of x:

∫ logb(x) dx = x ∙ ( logb(x) - 1 / ln(b) ) + C

For example:

∫ log2(x) dx = x ∙ ( log2(x) - 1 / ln(2) ) + C

Logarithm approximation

log2(x) ≈ n + (x/2n - 1) ,

Complex logarithm

For complex number z:

z = reiθ = x + iy

The complex logarithm will be (n = ...-2,-1,0,1,2,...):

Log z = ln(r) + i(θ+2nπ) = ln(√(x2+y2)) + i·arctan(y/x))

Logarithm problems and answers

Problem #1

Find x for

log2(x) + log2(x-3) = 2

Solution:

Using the product rule:

log2(x∙(x-3)) = 2

Changing the logarithm form according to the logarithm definition:

x∙(x-3) = 22

Or

x2-3x-4 = 0

Solving the quadratic equation:

x1,2 = [3±√(9+16) ] / 2 = [3±5] / 2 = 4,-1

Since the logarithm is not defined for negative numbers, the answer is:

x = 4

Problem #2

Find x for

log3(x+2) - log3(x) = 2

Solution:

Using the quotient rule:

log3((x+2) / x) = 2

Changing the logarithm form according to the logarithm definition:

(x+2)/x = 32

Or

x+2 = 9x

Or

8x = 2

Or

x = 0.25

Graph of log(x)

log(x) is not defined for real non positive values of x:

Logarithms table

x log10x log2x logex
0 undefined undefined undefined
0+ - ∞ - ∞ - ∞
0.0001 -4 -13.287712 -9.210340
0.001 -3 -9.965784 -6.907755
0.01 -2 -6.643856 -4.605170
0.1 -1 -3.321928 -2.302585
1 0 0 0
2 0.301030 1 0.693147
3 0.477121 1.584963 1.098612
4 0.602060 2 1.386294
5 0.698970 2.321928 1.609438
6 0.778151 2.584963 1.791759
7 0.845098 2.807355 1.945910
8 0.903090 3 2.079442
9 0.954243 3.169925 2.197225
10 1 3.321928 2.302585
20 1.301030 4.321928 2.995732
30 1.477121 4.906891 3.401197
40 1.602060 5.321928 3.688879
50 1.698970 5.643856 3.912023
60 1.778151 5.906991 4.094345
70 1.845098 6.129283 4.248495
80 1.903090 6.321928 4.382027
90 1.954243 6.491853 4.499810
100 2 6.643856 4.605170
200 2.301030 7.643856 5.298317
300 2.477121 8.228819 5.703782
400 2.602060 8.643856 5.991465
500 2.698970 8.965784 6.214608
600 2.778151 9.228819 6.396930
700 2.845098 9.451211 6.551080
800 2.903090 9.643856 6.684612
900 2.954243 9.813781 6.802395
1000 3 9.965784 6.907755
10000 4 13.287712 9.210340

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