# Logarithm Calculator

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

Log Calculator log
logb(x) = y

b: log base number, b>0 and b≠1 / x: is real number, x>0

logb(x) = y, and x = logb(bx)
logb(x) = y and x = by

## 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:

 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:

logb 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.

 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 logb(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 BaseLog NameNotationLog Example
2binary logarithmlb(x)log2(16) = lb(16) = 4 => 24 = 16
10common logarithmlg(x)log10(1000) = lg(1000) = 3 => 103 = 1000
enatural logarithmln(x)loge(8) = ln(8) = 2.0794 => e2.0794 = 8

### Logarithmic Identities

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

#### Logarithm of a Product

```logb(x·y) = logb(x) + logb(y)
log2(5·7) = log2(5) + log2(7)```

#### Logarithm of a Quotient

```logb(x/y) = logb(x) - logb(y)
log2(5/7) = log2(5) - log2(7)```

#### Logarithm of a Power

```logb(xy) = y·logb(x)
log2(57) = 7·log2(5)```

#### Change of Base

```logb(x) = (logk(x)) / (logk(b))
```
##### Natural Logarithm Examples
• ln(2) = loge(2) = 0.6931
• ln(3) = loge(3) = 1.0986
• ln(4) = loge(4) = 1.3862
• ln(5) = loge(5) = 1.609
• ln(6) = loge(6) = 1.7917
• ln(10) = loge(10) = 2.3025

#### Logarithm Values Tables

List of log function values tables in common base numbers.

log2(x)NotationValue
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)NotationValue
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
loge(x)NotationValue
loge(1)ln(1)0
loge(2)ln(2)0.693147
loge(3)ln(3)1.098612
loge(4)ln(4)1.386294
loge(5)ln(5)1.609438
loge(6)ln(6)1.791759
loge(7)ln(7)1.94591
loge(8)ln(8)2.079442
loge(9)ln(9)2.197225
loge(10)ln(10)2.302585
loge(11)ln(11)2.397895
loge(12)ln(12)2.484907
loge(13)ln(13)2.564949
loge(14)ln(14)2.639057
loge(15)ln(15)2.70805
loge(16)ln(16)2.772589
loge(17)ln(17)2.833213
loge(18)ln(18)2.890372
loge(19)ln(19)2.944439
loge(20)ln(20)2.995732
loge(21)ln(21)3.044522
loge(22)ln(22)3.091042
loge(23)ln(23)3.135494
loge(24)ln(24)3.178054