Logarithm Calculator

An online complete log logarithm calculator.

log

ln

log10

log2

Antilog

Logarithm is the inverse function to exponentiation.

First, we see about 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