# Logarithm Calculator

An online complete log logarithm calculator.

log

ln

log10

log2

Antilog

Logarithm is the inverse function to exponentiation.

Exponentiation is an expression that involves two numbers, a base and an exponent, where an exponent is mathematical shorthand representing how many times a number is multiplied against itself.

Example

21 = 2

22 = 4

23 = 8

24 = 16

25 = 32

25 = 64

with same above entries, Now we see about Logarithm, i.e Inverse function to exponentiation

log2 2 = 1

log2 4 = 2

log2 8 = 3

log2 16 = 4

log2 32 = 5

log2 64 = 6

###### Logarithm Formulas Used In Our Calculators

1. Log Formula

logb(x) = y

x = logb(bx)

logb(x) = y is equivalent to x = by

(logb(x) => This read as log base b of x is equals to y )

b: log base number, b>0 and b≠1

x: is real number, x>0

2. Natural Logarithm or Logarithm Base e Formula

Log base e is also called as natural logarithm.

Natural logarithm symbol is ln.

ln(x) = y

ln(x) is equivalent to loge(x)

x: is real number, x > 0

3. Common Logarithm or Logarithm Base 10 Formula

Log base 10 is also called as common logarithm.

log10(x) = y is equivalent to x = 10y

log10(x) = log(x)

x: is real number, x>0

4. Binary Logarithm or Logarithm Base 2 Formula

Log base 2 is also called as binary logarithm.

log2(x) = y is equivalent to x = 2y

x: is real number, x>0

5. Antilogarithm (or Inverse logarithm) Formula

Calculate the inverse logarithm of a number.

When

y = logb x

The anti-logarithm is calculated by raising the base b to the logarithm y

x = logb-1(y) = b y

## List Of Logarithmic Laws or Rules or Identities

Important formulas, sometimes called logarithmic identities or logarithmic laws.They are

Logarithm of a Product

logb (xy) = logb x + logb y

Example

log3 243 = log3 (9 . 27) = log3 9 + log3 27 = 2 + 3 = 5

Logarithm of a Quotient

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

Example

log2 16 = log2 (64/4) = log2 64 - log2 4 = 6 - 2 = 4

Logarithm of a Power

logb (xp) = p logb x

The logarithm of an power number where its base is the same as the base of the log equals the power.

Example

log2 64 = log2 (26) = 6 log2 2 = 6

Logarithm of a Root

logb  px = (logb x) / p

Example

log101000 = (1 / 2) . log10 1000 = 3 / 2 = 1.5

Logarithm of Zero

logb (1) = 0

The logarithm of 1 with b > 1 equals zero.

Logarithm of Identity

logb (b) = 1

The logarithm of a number that is equal to its base is just 1.

Logarithm of Exponent

blogb (k) = k

Raising the logarithm of a number by its base equals the number.

Change of Base

logb (x) = (logk (x)) / (logk (b))

## Common Values for Log Base b

Base bName for logbxISO notationOther notations
2Binary logarithmlb xld x, log x, lg x, log2x
eNatural logarithmln xlog x
10Common logarithmlg xlog x, log10x

## Logarithm Values Tables

logb(x) = y
log2 (1) = 0
log2 (2) = 1
log2 (3) = 1.584962501
log2 (4) = 2
log2 (5) = 2.321928095
log2 (6) = 2.584962501
log2 (7) = 2.807354922
log2 (8) = 3
log2 (9) = 3.169925001
log2 (10) = 3.321928095
log2 (11) = 3.459431619
log2 (12) = 3.584962501
log2 (13) = 3.700439718
log2 (14) = 3.807354922
log2 (15) = 3.906890596
log2 (16) = 4
log2 (17) = 4.087462841
log2 (18) = 4.169925001
log2 (19) = 4.247927513
log2 (20) = 4.321928095
log2 (21) = 4.392317423
log2 (22) = 4.459431619
log2 (23) = 4.523561956
log2 (24) = 4.584962501
log2 (25) = 4.64385619
log2 (26) = 4.700439718
log2 (27) = 4.754887502
log2 (28) = 4.807354922
log2 (29) = 4.857980995
log2 (30) = 4.906890596
log2 (31) = 4.95419631
log2 (32) = 5
log2 (33) = 5.044394119
log2 (34) = 5.087462841
log2 (35) = 5.129283017
log2 (36) = 5.169925001
log2 (37) = 5.209453366
log2 (38) = 5.247927513
log2 (39) = 5.285402219
log2 (40) = 5.321928095
log2 (41) = 5.357552005
log2 (42) = 5.392317423
log2 (43) = 5.426264755
log2 (44) = 5.459431619
log2 (45) = 5.491853096
log2 (46) = 5.523561956
log2 (47) = 5.554588852
log2 (48) = 5.584962501
log2 (49) = 5.614709844
log2 (50) = 5.64385619