Change of Base Formula

About Change of Base Formula

As the name says, the change of base formula is used to change the base of a logarithm. You may have noticed that there are just two buttons on a scientific calculator: "log" and "ln." We also know that "log" represents for a base 10 logarithm and "ln" stands for a base e logarithm. However, calculating the logarithm of a number in bases other than 10 and e is not an option.

The base change formula solves the problem of changing the base from e to 10 and back again. It is utilised to solve a variety of logarithms difficulties.

What Is Change of Base Formula?

The change of base formula is used to represent a logarithm of a number with a specific base as the ratio of two logarithms, each with a different base than the original logarithm. This is a logarithmic property.

Change of Base Formula

The following is a representation of changing the base formula. The base of the provided logarithm is also modified to a new base logarithm. The fundamental logarithm with a base is split into two logarithms with new and identical bases. The following is the new base formula:

logba = [logca] / [logcb]

In this, The argument of the numerator logarithm is the same as the original logarithm argument.

The base of the original logarithm is the same as the argument of the logarithm in the denominator.

Both the numerator and denominator logarithms should have the same base, which might be any positive number other than 1.

Note:Another form of this formula is, logbba ·logccb= logcca, which is also widely used in solving the problems.

Change of Base Formula Derivation

Change of base formula derivation can be understood from the following steps. Let us assume that

logbba = p,logcca = q, and logccb = r.

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

a = bp, a = cq, and b = cr.

From the first two equations,

bp= cq

Substituting b = cr(which is from third equation) here,

(cr)p= cq

crp= cq(Using a property of exponents, (am)n= amn)

pr = q

p = q / r

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

logbba = [logcca] / [logccb]

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Change of Base Formula

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