Change of Base Formula

The change of base formula is used to calculate a logarithm with a different base.

The formula is:

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 expression is a logarithmic characteristic.

Derivation

Below is the derivation of the 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

• 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

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

Practice Questions

Q1. Simplify: log₂64 + log₄16

Q2. Solve for x: log₂​x = log₄​81

Q3. Solve: \log_3(x+2)=\frac{\log 64}{\log 4}

Q4. Solve for x: log₄x=log₂12

Q5. Simplify: \frac{\log_2 3}{\log_2 27}+\frac{\log_3 4}{\log_3 16}