Change of base rule for Logarithm
Last Updated :
23 Jul, 2025
The change of base formula is a useful concept in mathematics. That allows you to convert a logarithm from one base to another. Change of base formula in logarithm allows us to rewrite a logarithm with a different base.
It allows us to compute logarithms using calculators or computational tools that may only support logarithms with certain bases, typically base 10 (log10) or natural logarithms (ln) which are present in scientific calculators. So, instead of calculating the logarithm directly with the given base, we can use a different base and adjust the formula accordingly.
Formula for Base Change of Log
This formula expresses a logarithm of a number with a particular base as a ratio of two logarithms, each with a different base than the original logarithm. This is a logarithmic characteristic. The formula is given as:
logba = logca / logcb
or
logba . logcb = logca
Below is the derivation of Change of Base Formula.
If logba = p, logca = q and logcb = r.
Then,
a = bp, a = cq, and b = cr.
Also, bp = cq.
Substituting b = cr, we have:
⇒ (cr)p = cq
Using (am)n = amn
⇒ crp = cq
⇒ pr = q
p = q/r
Substituting the values of p, q, and r, we have:
logba = logca / log b.
Importance of Change of Base in Log
- The rule of base change is a logarithmic property that enables the input of a logarithm with a base other than 10 into a calculator.
- The equation is: logdc = log10c/log10d
- The base change formula is useful for calculating logarithms on a calculator that only supports base 10.
- The base change formula can also help simplify certain logarithmic expressions.
- The base change formula is compatible with the natural logarithm, ln:
log(b) = ln(b)/ln(a) . - Most calculators provide the option to input the base of a logarithm.
- The formula is only applicable to logarithms with positive bases.
- Both the numerator and the denominator of the formula represent logarithms with the same base c.
Question 1: Evaluate log648 using the change of base formula.
Solution:
log648 = {log 8}/{log 64}
⇒ log648 = log 8/ log 82
Using the property log am = m log a, we have:
⇒ log648 = log 8/ 2 log 8
⇒ log648 = 1/2
Question 2: Evaluate log119.
Solution:
Using the change of base formula, we have:
log119 = log 9/ log 11
= 0.95452/1.0413 = 0.91667
Question 3: Evaluate log98.
Solution:
Using the change of base formula, we have:
log98 = log 8/ log 9
= 0.90308/0.95424 = 0.9464
Question 4: Evaluate log1110.
Solution:
Using the change of base formula, we have:
log1110= log 10/ log 11
= 0.8655/0.57849 = 0.8755
Question 5: Evaluate log65.
Solution:
Using the change of base formula, we have:
log65 = log 5/ log 6
= 0.8982
Question 6: Evaluate log43.
Solution:
Using the change of base formula, we have:
log43 = log 3/ log 4
= 0.7924
Question 7: Evaluate log87.
Solution:
Using the change of base formula, we have:
log87 = log 7/ log 8
= 0.9357
Also read,
Similar Reads
Logarithm In mathematics, a logarithm is the inverse operation of exponentiation. It is defined as the power to which the base number must be raised to get the given number.Logarithms serve as mathematical tools that help simplify complex calculations involving exponential relationships. If you know that bx =
3 min read
Laws of Logarithms The logarithm is the exponent or power to which a base is raised to get a particular number. For example, 'a' is the logarithm of 'm' to the base of 'x' if xm = a, then we can write it as m = logxa. Logarithms are invented to speed up the calculations and time will be reduced when we are multiplying
4 min read
Log Rules Logarithm rules are used to simplify and work with logarithmic expressions. They help relate logarithms to exponents and make complex calculations easier.A logarithm is the inverse of an exponent. It answers the question: "To what power must a base be raised to get a certain number?"Out of all these
7 min read
Logarithm Formula Logarithm is defined as the power to which a number is raised to yield some other values. Logarithms are the inverse of exponents. There is a unique way of reading the logarithm expression. For example, bx = n is called as 'x is the logarithm of n to the base b.There are two parts of the logarithm:
6 min read
Continuity and Differentiability of Logarithmic Function The word continuity means something which is continuous in nature. The flow of water is continuous, time in real life is continuous, and many more instances show the continuity in real life. In mathematics, the Continuous function is the one which when drawn on a graph does not show any breaks and i
5 min read
Derivative of Logarithmic Functions Derivative or Differentiation of Logarithmic Function as the name suggests, explores the derivatives of log functions with respect to some variable. As we know, derivatives are the backbone of Calculus and help us solve various real-life problems. Derivatives of the log functions are used to solve v
10 min read
Logarithmic Differentiation Method of finding a function's derivative by first taking the logarithm and then differentiating is called logarithmic differentiation. This method is specially used when the function is type y = f(x)g(x). In this type of problem where y is a composite function, we first need to take a logarithm, ma
8 min read
Log Table A Log Table in math is a reference tool to ease computations using logarithmic functions. It usually provides pre-computed logarithm values for various integers, commonly in a base like 10 or the natural logarithm base (e.g., â 2.71828). Log tables allow users to obtain the logarithm of a given numb
9 min read
Antilog Table Antilog Table in math is a refrence tool used to reverse logarithmic computation and retrieve the original number from its logarithmic value.It typically provides a pre-computed antilogarithm value for various logarithmic inputs, commonly in base 1o or natural logarithm base( e.g â
2.71828)Antilog t
7 min read
Change of Base in Logarithim The change of base formula is a useful concept in mathematics. That allows you to convert a logarithm from one base to another. Change of base formula in logarithm allows us to rewrite a logarithm with a different base. It allows us to compute logarithms using calculators or computational tools that
4 min read
Difference Between Log and Ln Logarithms(log) and natural logarithms(ln) are fundamental mathematical concepts that simplify complex calculations involving exponential relationships Logarithms are essential for solving equations where an unknown variable appears as the exponent of some other quantity. A logarithm can have any po
3 min read