1. Introduction
Sometimes the base of a logarithm is not convenient to work with. For example, calculator buttons usually provide only two logs: log (base 10) and ln (base e).
The change of base formula helps convert a log from one base to another so that it can be evaluated easily.
2. Why Do We Need Change of Base?
We use the change of base formula when:
- The base is not 10 or e (calculator can’t compute it directly).
- We want to simplify expressions.
- We want to compare logarithms with different bases.
- We need a uniform base for solving equations.
3. The Change of Base Formula
The general formula for converting a logarithm from base \(b\) to base \(k\) is:
\( \log_b a = \dfrac{\log_k a}{\log_k b} \)
This works for any valid base \(k\) (10, e, 2, etc.).
3.1. Using Common Logarithm (base 10)
\( \log_b a = \dfrac{\log a}{\log b} \)
This is useful because calculators have a log button.
3.2. Using Natural Logarithm (base e)
\( \log_b a = \dfrac{\ln a}{\ln b} \)
This works the same way using the ln key.
4. Understanding the Formula (Simple Idea)
Every logarithm measures how many times a base must be multiplied to reach a certain number.
Changing the base does not change the value of the logarithm — it only changes the unit of measurement.
Similar to converting kilometres to metres, we convert logs from one base to another.
5. Worked Examples
These examples show how to use the change of base formula.
5.1. Example 1: Convert to base 10
Find \( \log_2 8 \).
Using change of base:
\( \log_2 8 = \dfrac{\log 8}{\log 2} = \dfrac{0.903}{0.301} = 3 \)
5.2. Example 2: Convert to natural log
Find \( \log_5 20 \).
\( \log_5 20 = \dfrac{\ln 20}{\ln 5} \approx \dfrac{2.9957}{1.6094} \approx 1.86 \)
5.3. Example 3: Using any base (e.g., base 3)
Find \( \log_7 49 \).
\( \log_7 49 = \dfrac{\log_3 49}{\log_3 7} \)
Since 49 = 7²:
\( \log_7 49 = 2 \)
6. Using Change of Base on Calculator
Most calculators do not have a key for log base 2, 5, or 7. Instead, you can compute:
- log(a) ÷ log(b)
- ln(a) ÷ ln(b)
Both methods give the same result.
7. Common Mistakes
- Writing \( \log_b a = \dfrac{\log b}{\log a} \) (incorrect order).
- Forgetting that the base stays in the denominator.
- Assuming the change of base changes the value (it doesn’t).
- Trying to take log of negative numbers (not defined).
8. Quick Practice
Use change of base to evaluate:
- \( \log_3 9 \)
- \( \log_2 5 \)
- \( \log_7 2 \)
- \( \log_{4} 64 \)
9. Summary
- The change of base formula converts any logarithm to a more convenient base.
- Formula: \( \log_b a = \dfrac{\log_k a}{\log_k b} \)
- Common choices: log (base 10) and ln (base e).
- Useful for simplifying expressions and using calculators.