# Logarithm Calculator

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

_{b}(x) = y

log_{b}(x) = y

x = log_{b}(b^{x})

**log _{b}(x) = y** is equivalent to

**x = b**

^{y}**b: ** log base number, b>0 and b≠1

**x: ** is real number, x>0

## 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:

2^{1} | 2^{2} | 2^{3} | 2^{4} | 2^{5} | 2^{6} |

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, 2^{3} = 8 ⇒ log_{2} 8 = 3 (the logarithm of 8 to base 2 is equal to 3, because 2^{3} = 8).

Similarly, log_{2} 64 = 6, because 2^{6} = 64.

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

2^{1} | 2^{2} | 2^{3} | 2^{4} | 2^{5} | 2^{6} |

2 | 4 | 8 | 16 | 32 | 64 |

log_{2}2 = 1 |
log_{2}4 = 2 |
log_{2}8 = 3 |
log_{2}16 = 4 |
log_{2}32 = 5 |
log_{2}64 = 6 |

Unfortunately, not all logarithms can be calculated that easily.
For example, finding log_{2} 5 is hardly possible by just using our simple calculation abilities.
After using logarithm calculator, we can find out that

log_{2} 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.

log_{2}(8) = 3 (log base 2 of 8) The exponential is 2^{3}= 8

### Common Values for Log Base

Log Base | Log Name | Notation | Log Example |
---|---|---|---|

2 | binary logarithm | lb(x) | log_{2}(16) = lb(16) = 4 => 2^{4} = 16 |

10 | common logarithm | lg(x) | log_{10}(1000) = lg(1000) = 3 => 10^{3} = 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) log_{2}(5·7) = log_{2}(5) + log_{2}(7)

#### Logarithm of a Quotient

log_{b}(x/y) = log_{b}(x) - log_{b}(y) log_{2}(5/7) = log_{2}(5) - log_{2}(7)

#### Logarithm of a Power

log_{b}(x^{y}) = y·log_{b}(x) log_{2}(5^{7}) = 7·log_{2}(5)

#### Change of Base

log_{b}(x) = (log_{k}(x)) / (log_{k}(b))

#### Logarithm Values Tables

List of log function values tables in common base numbers.

Number (x) | Notation | log_{2}(x) |
---|---|---|

log_{2}(1) | lb(1) | 0 |

log_{2}(2) | lb(2) | 1 |

log_{2}(3) | lb(3) | 1.584963 |

log_{2}(4) | lb(4) | 2 |

log_{2}(5) | lb(5) | 2.321928 |

log_{2}(6) | lb(6) | 2.584963 |

log_{2}(7) | lb(7) | 2.807355 |

log_{2}(8) | lb(8) | 3 |

log_{2}(9) | lb(9) | 3.169925 |

log_{2}(10) | lb(10) | 3.321928 |

log_{2}(11) | lb(11) | 3.459432 |

log_{2}(12) | lb(12) | 3.584963 |

log_{2}(13) | lb(13) | 3.70044 |

log_{2}(14) | lb(14) | 3.807355 |

log_{2}(15) | lb(15) | 3.906891 |

log_{2}(16) | lb(16) | 4 |

log_{2}(17) | lb(17) | 4.087463 |

log_{2}(18) | lb(18) | 4.169925 |

log_{2}(19) | lb(19) | 4.247928 |

log_{2}(20) | lb(20) | 4.321928 |

log_{2}(21) | lb(21) | 4.392317 |

log_{2}(22) | lb(22) | 4.459432 |

log_{2}(23) | lb(23) | 4.523562 |

log_{2}(24) | lb(24) | 4.584963 |

log_{2}(25) | lb(25) | 4.643856 |

log_{2}(26) | lb(26) | 4.70044 |

log_{2}(27) | lb(27) | 4.754888 |

log_{2}(28) | lb(28) | 4.807355 |

log_{2}(29) | lb(29) | 4.857981 |

log_{2}(30) | lb(30) | 4.906891 |

log_{2}(31) | lb(31) | 4.954196 |

log_{2}(32) | lb(32) | 5 |

log_{2}(33) | lb(33) | 5.044394 |

log_{2}(34) | lb(34) | 5.087463 |

log_{2}(35) | lb(35) | 5.129283 |

log_{2}(36) | lb(36) | 5.169925 |

log_{2}(37) | lb(37) | 5.209453 |

log_{2}(38) | lb(38) | 5.247928 |

log_{2}(39) | lb(39) | 5.285402 |

log_{2}(40) | lb(40) | 5.321928 |

log_{2}(41) | lb(41) | 5.357552 |

log_{2}(42) | lb(42) | 5.392317 |

log_{2}(43) | lb(43) | 5.426265 |

log_{2}(44) | lb(44) | 5.459432 |

log_{2}(45) | lb(45) | 5.491853 |

log_{2}(46) | lb(46) | 5.523562 |

log_{2}(47) | lb(47) | 5.554589 |

log_{2}(48) | lb(48) | 5.584963 |

log_{2}(49) | lb(49) | 5.61471 |

log_{2}(50) | lb(50) | 5.643856 |

log_{2}(51) | lb(51) | 5.672425 |

log_{2}(52) | lb(52) | 5.70044 |

log_{2}(53) | lb(53) | 5.72792 |

log_{2}(54) | lb(54) | 5.754888 |

log_{2}(55) | lb(55) | 5.78136 |

log_{2}(56) | lb(56) | 5.807355 |

log_{2}(57) | lb(57) | 5.83289 |

log_{2}(58) | lb(58) | 5.857981 |

log_{2}(59) | lb(59) | 5.882643 |

log_{2}(60) | lb(60) | 5.906891 |

log_{2}(61) | lb(61) | 5.930737 |

log_{2}(62) | lb(62) | 5.954196 |

log_{2}(63) | lb(63) | 5.97728 |

log_{2}(64) | lb(64) | 6 |

Number (x) | Notation | log_{10}(x) |
---|---|---|

log_{10}(1) | log(1) | 0 |

log_{10}(2) | log(2) | 0.30103 |

log_{10}(3) | log(3) | 0.477121 |

log_{10}(4) | log(4) | 0.60206 |

log_{10}(5) | log(5) | 0.69897 |

log_{10}(6) | log(6) | 0.778151 |

log_{10}(7) | log(7) | 0.845098 |

log_{10}(8) | log(8) | 0.90309 |

log_{10}(9) | log(9) | 0.954243 |

log_{10}(10) | log(10) | 1 |

log_{10}(11) | log(11) | 1.041393 |

log_{10}(12) | log(12) | 1.079181 |

log_{10}(13) | log(13) | 1.113943 |

log_{10}(14) | log(14) | 1.146128 |

log_{10}(15) | log(15) | 1.176091 |

log_{10}(16) | log(16) | 1.20412 |

log_{10}(17) | log(17) | 1.230449 |

log_{10}(18) | log(18) | 1.255273 |

log_{10}(19) | log(19) | 1.278754 |

log_{10}(20) | log(20) | 1.30103 |

log_{10}(21) | log(21) | 1.322219 |

log_{10}(22) | log(22) | 1.342423 |

log_{10}(23) | log(23) | 1.361728 |

log_{10}(24) | log(24) | 1.380211 |

log_{10}(25) | log(25) | 1.39794 |

log_{10}(26) | log(26) | 1.414973 |

log_{10}(27) | log(27) | 1.431364 |

log_{10}(28) | log(28) | 1.447158 |

log_{10}(29) | log(29) | 1.462398 |

log_{10}(30) | log(30) | 1.477121 |

log_{10}(31) | log(31) | 1.491362 |

log_{10}(32) | log(32) | 1.50515 |

log_{10}(33) | log(33) | 1.518514 |

log_{10}(34) | log(34) | 1.531479 |

log_{10}(35) | log(35) | 1.544068 |

log_{10}(36) | log(36) | 1.556303 |

log_{10}(37) | log(37) | 1.568202 |

log_{10}(38) | log(38) | 1.579784 |

log_{10}(39) | log(39) | 1.591065 |

log_{10}(40) | log(40) | 1.60206 |

log_{10}(41) | log(41) | 1.612784 |

log_{10}(42) | log(42) | 1.623249 |

log_{10}(43) | log(43) | 1.633468 |

log_{10}(44) | log(44) | 1.643453 |

log_{10}(45) | log(45) | 1.653213 |

log_{10}(46) | log(46) | 1.662758 |

log_{10}(47) | log(47) | 1.672098 |

log_{10}(48) | log(48) | 1.681241 |

log_{10}(49) | log(49) | 1.690196 |

log_{10}(50) | log(50) | 1.69897 |

log_{10}(51) | log(51) | 1.70757 |

log_{10}(52) | log(52) | 1.716003 |

log_{10}(53) | log(53) | 1.724276 |

log_{10}(54) | log(54) | 1.732394 |

log_{10}(55) | log(55) | 1.740363 |

log_{10}(56) | log(56) | 1.748188 |

log_{10}(57) | log(57) | 1.755875 |

log_{10}(58) | log(58) | 1.763428 |

log_{10}(59) | log(59) | 1.770852 |

log_{10}(60) | log(60) | 1.778151 |

log_{10}(61) | log(61) | 1.78533 |

log_{10}(62) | log(62) | 1.792392 |

log_{10}(63) | log(63) | 1.799341 |

log_{10}(64) | log(64) | 1.80618 |

Number (x) | Notation | ln(x) |
---|---|---|

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 |

log_{e}(25) | ln(25) | 3.218876 |

log_{e}(26) | ln(26) | 3.258097 |

log_{e}(27) | ln(27) | 3.295837 |

log_{e}(28) | ln(28) | 3.332205 |

log_{e}(29) | ln(29) | 3.367296 |

log_{e}(30) | ln(30) | 3.401197 |

log_{e}(31) | ln(31) | 3.433987 |

log_{e}(32) | ln(32) | 3.465736 |

log_{e}(33) | ln(33) | 3.496508 |

log_{e}(34) | ln(34) | 3.526361 |

log_{e}(35) | ln(35) | 3.555348 |

log_{e}(36) | ln(36) | 3.583519 |

log_{e}(37) | ln(37) | 3.610918 |

log_{e}(38) | ln(38) | 3.637586 |

log_{e}(39) | ln(39) | 3.663562 |

log_{e}(40) | ln(40) | 3.688879 |

log_{e}(41) | ln(41) | 3.713572 |

log_{e}(42) | ln(42) | 3.73767 |

log_{e}(43) | ln(43) | 3.7612 |

log_{e}(44) | ln(44) | 3.78419 |

log_{e}(45) | ln(45) | 3.806662 |

log_{e}(46) | ln(46) | 3.828641 |

log_{e}(47) | ln(47) | 3.850148 |

log_{e}(48) | ln(48) | 3.871201 |

log_{e}(49) | ln(49) | 3.89182 |

log_{e}(50) | ln(50) | 3.912023 |

log_{e}(51) | ln(51) | 3.931826 |

log_{e}(52) | ln(52) | 3.951244 |

log_{e}(53) | ln(53) | 3.970292 |

log_{e}(54) | ln(54) | 3.988984 |

log_{e}(55) | ln(55) | 4.007333 |

log_{e}(56) | ln(56) | 4.025352 |

log_{e}(57) | ln(57) | 4.043051 |

log_{e}(58) | ln(58) | 4.060443 |

log_{e}(59) | ln(59) | 4.077537 |

log_{e}(60) | ln(60) | 4.094345 |

log_{e}(61) | ln(61) | 4.110874 |

log_{e}(62) | ln(62) | 4.127134 |

log_{e}(63) | ln(63) | 4.143135 |

log_{e}(64) | ln(64) | 4.158883 |

##### Helpful Logarithm References

- Log functions (Lamar Edu)
- Logarithm (Utah Edu)

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