Mr Calcu | Easily solve logs with any base—fast, accurate, and beginner-friendly.

Calculate and convert logarithms effortlessly. Master any base with ease using this intuitive tool—explore your math potential and verify results fast.

Logarithm Calculator Description

What Is a Logarithm?

A logarithm answers the question: "To what power must a base number be raised to obtain a given value?"

• Example: log₂(8) = 3 because 2³ = 8
• General Form:

  logb(x) = y ⟺ b^y = x

• Logarithms are the inverse of exponentiation.

Change of Base Formula

Most calculators support only log base 10 (log) or base e (ln). To compute logarithms of other bases, use:

logb(x) = logc(x) / logc(b)

• Common bases: 10 and e (≈ 2.718)
• Useful in: algorithm design, signal processing, compound interest calculations

Derivation

logb(x) = y ⟺ b^y = x
logc(b^y) = logc(x)
y * logc(b) = logc(x)
y = logc(x) / logc(b)

Edge Case Explanations

• log₁₀(1) = 0: Any number raised to 0 is 1
• log_b(b) = 1: The base raised to 1 equals itself
• log_b(x) where x < 0: Undefined in the real number system
• log_b(0): Undefined; as x → 0⁺, log(x) → -∞
• log₁(x): Undefined; base must not equal 1

Real-World Case Studies

1. Finance – Compound Interest

If your investment doubles every 10 years, to grow 8x:

log2(8) = 3 → 3 × 10 = 30 years

2. Computer Science – Binary Search

To search 1024 items using binary search:

log2(1024) = 10 → 10 comparisons max

Try the calculator now and turn complex logs into clear answers in seconds!