Change of base rule for Logarithm (original) (raw)

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:

**log b a = log c a / log c b
or
**log b a . log c b = log c a

**Derivation of Change of Base Formula

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: **log d c = log 10 c/log 10 d
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.

**Solved Questions using Change of Base Formula

**Question 1: Evaluate log 64 8 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 log 11 9.
**Solution:

Using the change of base formula, we have:

log119 = log 9/ log 11
= 0.95452/1.0413 **= 0.91667

**Question 3: Evaluate log 9 8.
**Solution:

Using the change of base formula, we have:

log98 = log 8/ log 9
= 0.90308/0.95424 **= 0.9464

**Question 4: Evaluate log 11 10.
**Solution:

Using the change of base formula, we have:

log1110= log 10/ log 11
= 0.8655/0.57849 **= 0.8755

**Question 5: Evaluate log 6 5.
**Solution:

Using the change of base formula, we have:

log65 = log 5/ log 6
**= 0.8982

**Question 6: Evaluate log 4 3.
**Solution:

Using the change of base formula, we have:

log43 = log 3/ log 4
**= 0.7924

**Question 7: Evaluate log 8 7.
**Solution:

Using the change of base formula, we have:

log87 = log 7/ log 8
**= 0.9357

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