log formulas

Logarithm

A logarithm is simply a different way to express exponents in writing. There are times when the exponents won’t work, therefore we resort to logarithms. The rules of exponents are used to derive a variety of logarithm formulas. To better understand them, let’s look at a few solved examples.

Log formulas

Let’s review a few things before diving into the log formulas. If you don’t mention it, logarithms come in two varieties: the common logarithm (often known as “log”) and the natural logarithm (which is written as “ln” and its base is always “e”). The formulas for common logarithms are provided in the next section. They can, however, be applied to natural logarithms as well. Here are some of the most regularly used log formulas for your convenience.

logb 1 = 0

logb b = 1

logb (xy) = logb x + logb y

logb (x / y) = logb x – logb y

logb ax = x logb a

logba = (logc a) / (logc b)

Logarithmic Formulas Derivation

Some of the most essential log formulas may be found here. In order to derive logarithmic formulas, we turn to the exponentiation laws.

Product Formula of logarithms

The product formula of logs is, logb (xy) = logb x + logb y.

Let us assume that logb x = m and logb y = n. Then by the definition of the logarithm,

x = bm and y = bn

Then xy = bm × bn = bm + n (by a law of exponents, am × an = am + n

Changing xy = bm + n to logarithmic form, 

m + n = logb xy

Put logbx = m and logb y = n 

logb (xy) = logb x + logb y

Quotient Formula of logarithms

The quotient formula of logs is, logb (x/y) = logb x – logb y.

Derivation:

Let us assume that logb x = m and logb y = n. Then by the definition of the logarithm,

x = bm and y = bn 

Then x/y = bm / bn = bm – n (by a law of exponents, am / an = am – n)

Changing x/y = bm – n to logarithmic form, 

m – n = logb (x/y)

Put logb x = m and logb y = n here,

logb (x/y) = logb x – logb y

Power Formula of Logarithms

The power formula of logarithms is logb ax = x logb a.

Derivation:

Let logb a = m. Then by the definition of logarithm, a = bm.

Raising both sides by x, we get

ax  = (bm)x

ax = bmx (by a law of exponents, (am)n = amn)

Changing back to logarithmic form,

logb ax = m x

Put m = logb a 

logb ax = x logb a