The change of base formula is a useful concept in mathematics. Especially when dealing with logarithms, It 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. Instead of calculating the logarithm directly with the given base, we can use a different base and adjust the formula accordingly.

The significance of the change of Base Formula lies in its practical applications. 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⁡). This formula is also useful in solving equations involving logarithms, simplifying expressions, and proving various mathematical identities.

Change of Base Formula

The change of base formula is used to alter the base of a logarithm, as its name implies. The scientific calculator only has two buttons: log and ln, where the former represents a base 10 logarithm and the latter is used for a base e logarithm. But these buttons don’t calculate for values of bases other than 10 and e. This problem is resolved by changing the basic formula. It’s also utilized to solve a variety of logarithms difficulties.

Key Points on Change of Base

• Change base rule rewrites logarithms for any base using base-10.
• Useful for calculators limited to base-10 logarithms.
• Can simplify some logarithm expressions.
• Applies to positive bases only.
• Formula involves dividing logarithms with the same base (c) in both numerator and denominator.

Base Change Formula of Log

This formula is employed when expressing 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 

Hence proved.

Properties of Log Change of Base

• 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

Change of Base Formula – FAQs

How to use change of base formula?

• Identify the logarithm and its base.
• Choose a new base (commonly 10 or e).
• Apply the formula: log_a(b) = (log_c(b)) / (log_c(a)).
• Calculate the new base logarithms and simplify if needed.

When to use change of base formula?

• To evaluate logarithms with bases not supported by calculators or software.
• When solving equations with different bases.
• For comparing logarithmic functions with various bases.
• To simplify expressions or reveal patterns.

How do you change log base 2 to base e?

• Use the formula: log_a(b) = (log_c(b)) / (log_c(a)).
• Set a = 2 and c = e to change from base 2 to base e.
• Equation becomes: log₂(x) = (log_e(x)) / (log_e(2)).

How do you change log base e to log base 10?

• Use the formula: log_a(b) = (log_c(b)) / (log_c(a)).
• Set a = e and c = 10 to change from base e to base 10.
• Equation becomes: log_e(x) = (log₁₀(x)) / (log₁₀(e)).
• Since log₁₀(e) is a constant, you can simplify further if needed.