Logarithm Calculator

Log calculator finds the logarithm function result (can be called exponent) from the given base number and a real number.

Logarithm

Logarithm is considered to be one of the basic concepts in mathematics. There are plenty of definitions, starting from really complicated and ending up with rather simple ones. In order to answer a question, what a logarithm is, let's take a look at the table below:

log base 1	log base 2	log base 3	log base 4	log base 5	log base 6
2¹	2²	2³	2⁴	2⁵	2⁶
2	4	8	16	32	64

This is the table in which we can see the values of two squared, two cubed, and so on. This is an operation in mathematics, known as exponentiation. If we look at the numbers at the bottom line, we can try to find the power value to which 2 must be raised to get this number. For example, to get 16, it is necessary to raise two to the fourth power. And to get a 64, you need to raise two to the sixth power.

Therefore, logarithm is the exponent to which it is necessary to raise a fixed number (which is called the base), to get the number y. In other words, a logarithm can be represented as the following:

log_b x = y

with b being the base, x being a real number, and y being an exponent.

For example, 2³ = 8 ⇒ log₂ 8 = 3 (the logarithm of 8 to base 2 is equal to 3, because 2³ = 8).
Similarly, log₂ 64 = 6, because 2⁶ = 64.

Therefore, it is obvious that logarithm operation is an inverse one to exponentiation.

log base 1	log base 2	log base 3	log base 4	log base 5	log base 6
2¹	2²	2³	2⁴	2⁵	2⁶
2	4	8	16	32	64
log₂2 = 1	log₂4 = 2	log₂8 = 3	log₂16 = 4	log₂32 = 5	log₂64 = 6

Unfortunately, not all logarithms can be calculated that easily. For example, finding log₂ 5 is hardly possible by just using our simple calculation abilities. After using logarithm calculator, we can find out that

log₂ 5 = 2,32192809

There are a few specific types of logarithms. For example, the logarithm to base 2 is known as the binary logarithm, and it is widely used in computer science and programming languages. The logarithm to base 10 is usually referred to as the common logarithm, and it has a huge number of applications in engineering, scientific research, technology, etc. Finally, so called natural logarithm uses the number e (which is approximately equal to 2.71828) as its base, and this kind of logarithm has a great importance in mathematics, physics, and other precise sciences.

The logarithm log_b(x) = y is read as log base b of x is equals to y.
Please note that the base of log number b must be greater than 0 and must not be equal to 1. And the number (x) which we are calculating log base of (b) must be a positive real number.

For example log 2 of 8 is equal to 3.

log₂(8) = 3 (log base 2 of 8)
The exponential is 2³ = 8

Common Values for Log Base

Log Base	Log Name	Notation	Log Example
2	binary logarithm	lb(x)	log₂(16) = lb(16) = 4 => 2⁴ = 16
10	common logarithm	lg(x)	log₁₀(1000) = lg(1000) = 3 => 10³ = 1000
e	natural logarithm	ln(x)	log_e(8) = ln(8) = 2.0794 => e^2.0794 = 8

Logarithmic Identities

List of logarithmic identites, formulas and log examples in logarithm form.

Logarithm of a Product

log_b(x·y) = log_b(x) + log_b(y)
log₂(5·7) = log₂(5) + log₂(7)

Logarithm of a Quotient

log_b(x/y) = log_b(x) - log_b(y)
log₂(5/7) = log₂(5) - log₂(7)

Logarithm of a Power

log_b(x^y) = y·log_b(x)
log₂(5⁷) = 7·log₂(5)

Change of Base

log_b(x) = (log_k(x)) / (log_k(b))

Natural Logarithm Examples

ln(2) = log_e(2) = 0.6931
ln(3) = log_e(3) = 1.0986
ln(4) = log_e(4) = 1.3862
ln(5) = log_e(5) = 1.609
ln(6) = log_e(6) = 1.7917
ln(10) = log_e(10) = 2.3025

Logarithm Values Tables

List of log function values tables in common base numbers.

log₂(x)	Notation	Value
log₂(1)	lb(1)	0
log₂(2)	lb(2)	1
log₂(3)	lb(3)	1.584963
log₂(4)	lb(4)	2
log₂(5)	lb(5)	2.321928
log₂(6)	lb(6)	2.584963
log₂(7)	lb(7)	2.807355
log₂(8)	lb(8)	3
log₂(9)	lb(9)	3.169925
log₂(10)	lb(10)	3.321928
log₂(11)	lb(11)	3.459432
log₂(12)	lb(12)	3.584963
log₂(13)	lb(13)	3.70044
log₂(14)	lb(14)	3.807355
log₂(15)	lb(15)	3.906891
log₂(16)	lb(16)	4
log₂(17)	lb(17)	4.087463
log₂(18)	lb(18)	4.169925
log₂(19)	lb(19)	4.247928
log₂(20)	lb(20)	4.321928
log₂(21)	lb(21)	4.392317
log₂(22)	lb(22)	4.459432
log₂(23)	lb(23)	4.523562
log₂(24)	lb(24)	4.584963

log₁₀(x)	Notation	Value
log₁₀(1)	log(1)	0
log₁₀(2)	log(2)	0.30103
log₁₀(3)	log(3)	0.477121
log₁₀(4)	log(4)	0.60206
log₁₀(5)	log(5)	0.69897
log₁₀(6)	log(6)	0.778151
log₁₀(7)	log(7)	0.845098
log₁₀(8)	log(8)	0.90309
log₁₀(9)	log(9)	0.954243
log₁₀(10)	log(10)	1
log₁₀(11)	log(11)	1.041393
log₁₀(12)	log(12)	1.079181
log₁₀(13)	log(13)	1.113943
log₁₀(14)	log(14)	1.146128
log₁₀(15)	log(15)	1.176091
log₁₀(16)	log(16)	1.20412
log₁₀(17)	log(17)	1.230449
log₁₀(18)	log(18)	1.255273
log₁₀(19)	log(19)	1.278754
log₁₀(20)	log(20)	1.30103
log₁₀(21)	log(21)	1.322219
log₁₀(22)	log(22)	1.342423
log₁₀(23)	log(23)	1.361728
log₁₀(24)	log(24)	1.380211

log_e(x)	Notation	Value
log_e(1)	ln(1)	0
log_e(2)	ln(2)	0.693147
log_e(3)	ln(3)	1.098612
log_e(4)	ln(4)	1.386294
log_e(5)	ln(5)	1.609438
log_e(6)	ln(6)	1.791759
log_e(7)	ln(7)	1.94591
log_e(8)	ln(8)	2.079442
log_e(9)	ln(9)	2.197225
log_e(10)	ln(10)	2.302585
log_e(11)	ln(11)	2.397895
log_e(12)	ln(12)	2.484907
log_e(13)	ln(13)	2.564949
log_e(14)	ln(14)	2.639057
log_e(15)	ln(15)	2.70805
log_e(16)	ln(16)	2.772589
log_e(17)	ln(17)	2.833213
log_e(18)	ln(18)	2.890372
log_e(19)	ln(19)	2.944439
log_e(20)	ln(20)	2.995732
log_e(21)	ln(21)	3.044522
log_e(22)	ln(22)	3.091042
log_e(23)	ln(23)	3.135494
log_e(24)	ln(24)	3.178054