The change of base formula, as its name implies, is used to change the base of a logarithm. We might have seen that a scientific calculator has only “log” and “ln” buttons. Also, we acknowledge that “log” stands for a logarithm of base 10 and “ln” stands for a logarithm of base e.

But there is no option to determine the logarithm of a number with any other bases other than 10 and e. The change of base formula resolves this issue. Also, it is used in solving several logarithm problems. Let us read the change of base formula along with its proof and a few solved cases.

What Is Change of Base Formula? 

The change of base formula is utilized to write a logarithm of a number with a supplied base as the ratio of two logarithms each with the same base that is distinct from the base of the original logarithm. This is a property of logarithms. You can see the change of base formula hither.

Change of Base Formula 

The change of base formula is:

logbb a = [logcc a] / [logcc b]

In this formula,

• The argument of the logarithm in the numerator is similar to the argument of the original logarithm.
• The argument of the logarithm in the denominator is similar to the base of the original logarithm.
• The bases of both logarithms of the numerator and denominator should be the same and this base can be all positive numbers other than 1.

Note: Another form of this formula is, logbb a · logcc b = logcc a, which is also widely used in solving problems.

Change of Base Formula Derivation

Let us assume that

logbb a = p,logcc a = q, and logcc b = r.

By turning each of these into exponential form, we get,

a = b^p, a = c^q, and b = c^r.

From the first two equations,

b^p = c^q

Substituting b = c^r (which is from the third equation) here,

(c^r)^p = c^q

c^rp = c^q (Using a property of exponents, (a^m)ⁿ = a^mn)

pr = q

p = q / r

Substituting the values of p, q, and r here,

logbb a = [logcc a] / [logcc b]

FAQs

What Are the Applications of Change of Base Formula?

The change of base formula is essentially used to change the base of a logarithm to any desired base. This is many utilized to calculate the logarithms with any other base than 10 and “e” because the calculator has options to calculate the logarithms with bases 10 (log button) and e (ln button) only.

How to Use Change of Base Formula?

The change of base formula says logbb a = [logcc a] / [logcc b]. It means to change the base of a logarithm logbb a, we just use division [log a] / [log b] where these logarithms can have any (same) positive number as a bse.