Online Logarithm Calculator with Custom Base

James is preparing for his college entrance exam. His teacher asks:

“What’s the logarithm of 64 with base 2?”

James remembers logs have something to do with exponents, but the numbers are spinning in his head. Instead of wasting time, he tries an online logarithm calculator with a custom base.

Within seconds, he learns:

log₂(64) = 6

Because 2⁶ = 64.

This is exactly why such a calculator exists — to save time, clear confusion, and give accurate answers for any base.

What Is This Calculator?

The logarithm calculator with custom base finds the logarithm of any number a with respect to any base b.

In simple words: it answers the question —

“To what power must I raise b to get a?”

Example:

log₁₀(1000) = 3 → because 10³ = 1000  

Why Is It Important?

• Math & Exams: Essential in algebra, calculus, and competitive exams.
  
• Science & Engineering: Used in pH calculations, sound intensity (decibels), population growth, etc.
  
• Data Science & Computer Science: Basis of algorithms, complexity analysis (Big-O with log terms).
  
• Everyday Use: Helps in financial growth models, interest, and exponential changes.
  

Without logarithms, many real-world problems would be nearly impossible to solve manually.

Formula

The logarithm formula is:

log_b(a) = x  →  b^x = a

👉 Where:

• a = the number (must be > 0)
  
• b = the base (must be > 0 and ≠ 1)
  
• x = result of the logarithm
  

If your calculator only supports log base 10 or e, you can use the change of base formula:

log_b(a) = log(a) / log(b)

Step-by-Step: How This Calculator Works

1. Enter the number (a): This is the value you want the log of. Example: 64.
   
2. Enter the base (b): Choose any valid base (greater than 0, not equal to 1). Example: 2.
   
3. Click calculate.
   
4. See the result instantly. Example: log₂(64) = 6.
   
5. Check with exponents (optional): Confirm by raising the base to the result → 2⁶ = 64 ✅.
   

Common Bases You Should Know

• Base 10 (Common Log): log₁₀ — used in sciences.
  
• Base e (Natural Log): ln(x) = logₑ(x) — used in calculus and growth/decay problems.
  

Base 2 (Binary Log): log₂ — used in computer science, coding, and digital systems.

FAQ