Change of Base Formula Calculator
Convert logarithms from one base to another using the change of base formula
Common choices: 10 (common log) or e ≈ 2.718 (natural log)
How to Use This Calculator
Enter the Number
Input the number (x) for which you want to find the logarithm. This is the argument of the logarithm function.
Enter Original Base
Input the base (b) of the original logarithm you want to convert from. For example, if converting log₂(8), enter 2.
Enter New Base
Specify the base you want to convert to. Common choices are 10 (common logarithm) or e ≈ 2.71828 (natural logarithm). Default is 10.
Get Result
Click "Calculate" to see the logarithm converted to the new base using the change of base formula.
Formula
logb(x) = logc(x) / logc(b)
Where:
- x = the number (argument of the logarithm)
- b = original base
- c = new base
Example 1: Convert log₂(8) to base 10
log₂(8) = log₁₀(8) / log₁₀(2)
log₁₀(8) ≈ 0.9031
log₁₀(2) ≈ 0.3010
log₂(8) = 0.9031 / 0.3010 = 3
Example 2: Convert log₅(25) to base e (natural log)
log₅(25) = ln(25) / ln(5)
ln(25) ≈ 3.2189
ln(5) ≈ 1.6094
log₅(25) = 3.2189 / 1.6094 = 2
Example 3: Convert log₃(27) to base 10
log₃(27) = log₁₀(27) / log₁₀(3)
log₁₀(27) ≈ 1.4314
log₁₀(3) ≈ 0.4771
log₃(27) = 1.4314 / 0.4771 ≈ 3
About Change of Base Formula Calculator
The Change of Base Formula Calculator helps you convert logarithms from one base to another. This is essential when your calculator only has buttons for common logarithm (base 10) or natural logarithm (base e), but you need to calculate a logarithm with a different base. The change of base formula allows you to express any logarithm in terms of any other base.
When to Use This Calculator
- Calculator Limitations: When your calculator only has log₁₀ or ln buttons, but you need log₂, log₅, etc.
- Mathematical Problems: Solving logarithmic equations that require base conversion
- Computer Science: Converting between binary (base 2) and decimal (base 10) logarithms
- Mathematical Analysis: Simplifying logarithmic expressions by converting to a common base
- Educational Purposes: Understanding the relationship between logarithms of different bases
Why Use Our Calculator?
- ✅ Instant Conversion: Convert logarithms between any bases instantly
- ✅ Flexible: Supports any base conversion you need
- ✅ Accurate: Precise calculations with high decimal precision
- ✅ Educational: Shows the formula being used for learning purposes
- ✅ 100% Free: No registration or payment required
- ✅ Step-by-Step: See how the formula is applied in your calculation
Common Applications
Calculators: Most calculators only have log₁₀ and ln buttons. To calculate log₂(16), you can use: log₂(16) = log₁₀(16) / log₁₀(2) = 1.2041 / 0.3010 = 4.
Computer Science: Binary logarithms (log₂) are common in algorithm analysis. You can convert them to natural logarithms: log₂(n) = ln(n) / ln(2).
Mathematical Simplification: When solving logarithmic equations, converting all terms to the same base makes the problem easier to solve.
Tips for Best Results
- The new base can be any positive number except 1
- Common choices for new base are 10 (common log) or e ≈ 2.71828 (natural log)
- The number (x) must be positive
- Both bases must be positive and not equal to 1
- This formula works for any bases because logarithms with different bases are proportional
Frequently Asked Questions
Why do I need the change of base formula?
Most calculators only have buttons for common logarithm (base 10) and natural logarithm (base e). To calculate logarithms with other bases (like base 2, 5, or 7), you need the change of base formula.
Can I convert to any base?
Yes! You can convert a logarithm from any base to any other base, as long as both bases are positive numbers not equal to 1. The formula works universally.
What's the most common base conversion?
The most common conversions are from any base to base 10 (common logarithm) or base e (natural logarithm), because these are what most calculators support.
Does it matter which base I convert to?
No, mathematically it doesn't matter. However, base 10 and base e are most convenient because calculators typically have these functions built-in, making the intermediate calculations easier.
Can I use this formula in reverse?
Yes! The formula works both ways. If you have log₁₀(x), you can convert it to log₂(x) using: log₂(x) = log₁₀(x) / log₁₀(2).