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Logarithm

What is Log?

The logarithm, or log, is the inverse of the mathematical operation of exponentiation. This means that the log of a number is the number that a fixed base has to be raised to in order to yield the number. Conventionally, log implies that base 10 is being used, though the base can technically be anything. When the base is e, ln is usually written, rather than loge. log2, the binary logarithm, is another base that is typically used with logarithms. If for example:

x = by; then y = logbx; where b is the base

Each of the mentioned bases are typically used in different applications. Base 10 is commonly used in science and engineering, base e in math and physics, and base 2 in computer science.

Basic Log Rules

When the argument of a logarithm is the product of two numerals, the logarithm can be re-written as the addition of the logarithm of each of the numerals.

logb(x × y) = logbx + logby
EX: log(1 × 10) = log(1) + log(10) = 0 + 1 = 1

When the argument of a logarithm is a fraction, the logarithm can be re-written as the subtraction of the logarithm of the numerator minus the logarithm of the denominator.

logb(x / y) = logbx - logby
EX: log(10 / 2) = log(10) - log(2) = 1 - 0.301 = 0.699

If there is an exponent in the argument of a logarithm, the exponent can be pulled out of the logarithm and multiplied.

logbxy = y × logbx
EX: log(26) = 6 × log(2) = 1.806

It is also possible to change the base of the logarithm using the following rule.

logb(x) = 
logk(x)

logk(b)
EX: log10(x) = 
log2(x)

log2(10)

To switch the base and argument, use the following rule.

logb(c) = 
1

logc(b)
EX:   log5(2) = 
1

log2(5)

Other common logarithms to take note of include:

logb(1) = 0
logb(b) = 1
logb(0) = undefined
limx→0+logb(x) = - ∞
ln(ex) = x

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