Change of base rule for Logarithm

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_ba = log_ca / log_cb 
or
log_ba . log_cb = log_ca

Derivation of Change of Base Formula

Below is the derivation of Change of Base Formula.

If log_ba = p, log_ca = q and log_cb = r.

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

Also, b^p = c^q.

Substituting b = c^r, we have:
⇒ (c^r)^p = c^q

Using (a^m)ⁿ = a^mn
⇒ c^rp = c^q
⇒ pr = q

p = q/r

Substituting the values of p, q, and r, we have:
log_ba = log_ca / 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​/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

Also read,

• Logarithms - Complete tutorial
• Difference between Logarithm (log) and Natural log( ln)
• Logarithm Formulas
• Properties of Logarithms