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Change of base rule for Logarithm

Last Updated : 23 Jul, 2025
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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

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: logd​c = log10​c​/log10​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 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

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