Logarithm

Logarithm is mathematical function that find the power that we need to raise the base in order to get x.

• Logarithm definition
• Logarithm rules
• Logarithm table
• Logarithm Calculator

Logarithm definition

When:

b ^y = x

Then base b logarithm of x is:

log_b(x) = y

So the base b logarithm of x is the exponent that b should be raised y to get x.

Example

when

2⁴ = 16

Then

log₂(16) = 4

Logarithm as inverse function of exponential function

The logarithmic function log_b(x) is the inverse function of the exponential function b^x .

For x>0,

f (f ^-1(x)) = b^logb^(x) = x

Or

f ^-1(f (x)) = log_b(b^x) = x

Natural logarithm (ln)

Natural logarithm is a logarithm to the base e:

ln(x) = log_e(x)

When e constant is the number:

 

See: Natural logarithm

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

Logarithmic function

The logarithmic function has the basic form of:

f (x) = log_b(x)

Logarithm rules

Rule name	Rule
Logarithm product rule	logb(x ∙ y) = logb(x) + logb(y)
Logarithm quotient rule	logb(x / y) = logb(x) - logb(y)
Logarithm power rule	logb(x y) = y ∙ logb(x)
Logarithm base switch rule	logb(c) = 1 / logc(b)
Logarithm base change rule	logb(x) = logc(x) / logc(b)
Derivative of logarithm	f (x) = logb(x) ⇒ f ' (x) = 1 / ( x ln(b) )
Integral of logarithm	∫ logb(x) dx = x ∙ ( logb(x) - 1 / ln(b) ) + C
Logarithm of 0	logb(0) is undefined
Logarithm of 1	logb(1) = 0

See: Logarithm rules

 

Logarithm of product

The logarithm of a product of x and y is the sum of logarithm of x and logarithm of y.

log_b(x ∙ y) = log_b(x) + log_b(y)

For example:

log_b(3 ∙ 7) = log_b(3) + log_b(7)

Logarithm of ratio

The logarithm of a quotient of x and y is the difference of logarithm of x and logarithm of y.

log_b(x / y) = log_b(x) - log_b(y)

For example:

log_b(3 / 7) = log_b(3) - log_b(7)

Logarithm of power

log_b(x ^y) = y ∙ log_b(x)

For example:

log_b(2⁸) = 8 ∙ log_b(2)

Logarithm base switch

log_b(c) = 1 / log_c(b)

For example:

log₂(8) = 1 / log₈(2)

Logarithm base change

log_b(x) = log_c(x) / log_c(b)

Logarithm of 0

log_b(0)

The natural logarithm of zero is undefined, but the limit near 0 is minus infinity:

Logarithm of 1

log_b(1) = 0

For example:

log₂(1) = 0

Logarithm of the base

log_b(b) = 1

For example:

log₂(2) = 1

Logarithm derivative

When

f (x) = log_b(x)

Then the derivative of f(x):

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

Logarithm integral

The integral of logarithm of x:

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

Logarithm approximation

log₂(x) ≈ n + (x/2ⁿ - 1) ,

Logarithms table

x	log10 x	log2 x	loge x
0	undefined	undefined	undefined
0+	- ∞	- ∞	- ∞
0.0001	-4.000000	-13.287712	-9.210340
0.001	-3.000000	-9.965784	-6.907755
0.01	-2.000000	-6.643856	-4.605170
0.1	-1.000000	-3.321928	-2.302585
1	0.000000	0.000000	0.000000
2	0.301030	1.000000	0.693147
3	0.477121	1.584963	1.098612
4	0.602060	2.000000	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	3.000000	2.079442
9	0.954243	3.169925	2.197225
10	1.000000	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	2.000000	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	3.000000	9.965784	6.907755
10000	4.000000	13.287712	9.210340

 

Logarithm calculator ►

 

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See also

• Logarithm of zero
• Logarithm of one
• Logarithm of infinity
• Logarithm of negative number
• Logarithm calculator
• Natural logarithm calculator
• Natural logarithm - ln x
• e constant
• Decibel (dB)