Change of Base Formula Calculator
Convert logarithms between different bases using the change of base formula
Convert Logarithm Base
Quick Examples
The number inside the logarithm
The current base of the logarithm
The target base for conversion
Conversion Result
Change of Base Formula:
log_b(x) = log_a(x) / log_a(b)
log_3(9) = log_27(9) / log_27(3)
Step-by-Step Calculation:
1. Original expression: log_27(9)
2. Target: Convert to base 3
3. Formula: log_b(x) = log_a(x) / log_a(b)
4. Substituting: log_3(9) = log_27(9) / log_27(3)
5. log_27(9) = 0.66666667
6. log_27(3) = 0.33333333
7. Result: 0.66666667 / 0.33333333 = 2
Verification:
Using change of base formula: 2
Direct calculation: 2
Difference: 0.00e+0
✅ Calculation verified!
Alternative Expressions
In base 10:
log₁₀(9) / log₁₀(3) = 2
In natural log:
ln(9) / ln(3) = 2
Example: Converting log₂₇(9) to Base 3
Problem
Convert log₂₇(9) to base 3
We want to find what power of 3 gives us 9, knowing that 27^x = 9
Solution
Formula: log₃(9) = log₂₇(9) / log₂₇(3)
Given: x = 9, a = 27, b = 3
Calculation: log₂₇(9) = ? (this is what we're converting)
Note: 3² = 9 and 3³ = 27
So: log₃(9) = 2 and log₃(27) = 3
Using change of base: log₂₇(9) = log₃(9) / log₃(27) = 2/3
Result: log₂₇(9) = 2/3 ≈ 0.6667
Change of Base Formula
Main Formula
log_b(x) = log_a(x) / log_a(b)
Convert from base a to base b
Using Common Log
log_b(x) = log₁₀(x) / log₁₀(b)
Convert any base using base 10
Using Natural Log
log_b(x) = ln(x) / ln(b)
Convert any base using natural log
Common Conversions
Base 2 to Base 10
log₁₀(x) = log₂(x) / log₂(10)
Base 10 to Natural
ln(x) = log₁₀(x) / log₁₀(e)
Natural to Base 2
log₂(x) = ln(x) / ln(2)
Quick Reference
Understanding the Change of Base Formula
What is the Change of Base Formula?
The change of base formula allows us to convert a logarithm from one base to another. This is particularly useful when we need to calculate logarithms with uncommon bases using calculators that only have common logarithm (base 10) or natural logarithm (base e) functions.
Why Do We Need It?
- •Most calculators only have log₁₀ and ln functions
- •Problems often involve uncommon bases
- •Simplifies complex logarithmic expressions
- •Enables mathematical analysis and comparison
Applications
Computer Science
Converting between binary, decimal, and hexadecimal logarithms
Engineering
Signal processing, decibel calculations, pH measurements
Mathematics
Solving exponential equations, calculus applications
Finance
Interest rate calculations, growth rate analysis
How the Formula Works
Step 1: Take log of argument in intermediate base
Step 2: Take log of new base in same intermediate base
Step 3: Divide result from Step 1 by result from Step 2
Mathematical Proof:
Let y = log_b(x), then b^y = x
Taking log_a of both sides: log_a(b^y) = log_a(x)
Using power rule: y·log_a(b) = log_a(x)
Therefore: y = log_a(x) / log_a(b)