Log Calculator

Calculate logarithms with any base

Solve logarithmic equations by finding the logarithm value, or determine unknown variables

CreatorFrank Zhao

What do you want to calculate?

Number (x)

The number for which you want to find the logarithm. Must be positive. Type "e" for Euler's number.

Base (b)

The base of the logarithm. Common bases: 10 (common log), 2 (binary), or type "e" for natural log (e ≈ 2.71828)

Logarithm (y)

← Calculating this

The result of the logarithm operation (the exponent). Type "e" for Euler's number.

What is a Logarithm?

A logarithm answers the question: "What power do I need to raise a base to, to get a certain number?"

If you've ever wondered "2 to what power gives me 8?" — that's a logarithm. The answer is 3, because 2³ = 8. We write this as log₂(8) = 3.

The relationship between logarithms and exponents:

log_b(x) = y ⟺ b^y = x

Logarithms are everywhere — from measuring earthquake intensity (Richter scale) to calculating compound interest, analyzing algorithms in computer science, and even determining how loud a sound is (decibels).

How to Use This Calculator

This calculator handles three scenarios. Just pick what you want to find:

📊 Find the Logarithm (y)

You have a number and a base, and want to find the log value.

Example: What is log₂(32)?

→ Enter Number (x) = 32, Base (b) = 2

→ Result: 5 (because 2⁵ = 32)

🔢 Find the Number (x)

You know the base and the log result, and want to find the original number.

Example: If log₁₀(x) = 3, what is x?

→ Enter Base (b) = 10, Logarithm (y) = 3

→ Result: 1000 (because 10³ = 1000)

📐 Find the Base (b)

You have a number and know the log result, and need to find the base.

Example: log_b(81) = 4. What is the base?

→ Enter Number (x) = 81, Logarithm (y) = 4

→ Result: 3 (because 3⁴ = 81)

💡 Pro tip: Type "e" in the base field to use the natural logarithm (base e ≈ 2.71828), commonly used in calculus, physics, and exponential growth problems.

Common Types of Logarithms

Common Log (base 10)

Written as log(x) or log₁₀(x)

Used in: pH calculations, decibels, Richter scale, scientific notation

Natural Log (base e)

Written as ln(x) or log_e(x)

Used in: Compound interest, population growth, radioactive decay, calculus

Binary Log (base 2)

Written as log₂(x) or lb(x)

Used in: Computer science, algorithm analysis, information theory, music (octaves)

Real-World Examples

🌍

Earthquake Magnitude

A magnitude 6 earthquake releases about 31.6 times more energy than a magnitude 5. That's because the Richter scale uses log₁₀. Each whole number increase means 10× more ground motion and roughly 31.6× more energy.

💰

Investment Doubling Time

Want to know when your investment doubles? With 7% annual return, it takes about ln(2) / ln(1.07) ≈ 10.24 years. This uses the natural log to solve exponential growth problems.

💻

Binary Search Efficiency

Searching through 1 million sorted items using binary search takes at most log₂(1,000,000) ≈ 20 comparisons. That's why programmers love logarithms — they turn huge problems into manageable ones.

Quick Reference: Log Properties

Handy rules for simplifying logarithmic expressions

Product Rule

log_b(xy) = log_b(x) + log_b(y)

Multiply inside → Add outside

Quotient Rule

log_b(x/y) = log_b(x) − log_b(y)

Divide inside → Subtract outside

Power Rule

log_b(xⁿ) = n · log_b(x)

Exponent inside → Multiply outside

Change of Base

log_b(x) = log_c(x) / log_c(b)

Convert any base using this formula

Log of 1

log_b(1) = 0

Any base, always zero (b⁰ = 1)

Log of Base

log_b(b) = 1

Base to itself, always one (b¹ = b)

Frequently Asked Questions

?Why can't I take the log of a negative number?

No positive base raised to any power gives a negative result. For example, 2¹ = 2, 2⁰ = 1, 2⁻¹ = 0.5 — always positive! In complex numbers it's possible, but this calculator works with real numbers only.

?What's special about base e (≈ 2.71828)?

The number e appears naturally in continuous growth — bacteria populations, radioactive decay, compound interest. Fun fact: Jacob Bernoulli discovered it in 1683 while studying compound interest!

?Why do calculators have LOG and LN buttons?

LOG = log₁₀ (common log, used in science)
LN = logₑ (natural log, used in calculus)
These two are so common they get dedicated buttons.

?What does log(1) always equal?

Zero — always, no matter the base. Why? Because any number raised to the power of 0 equals 1. So b⁰ = 1 means log_b(1) = 0. Try it yourself!

?How accurate are the results?

This calculator uses double-precision math with 15-17 significant digits of accuracy. More than enough for homework, engineering, or scientific research.

?Can I use this for homework?

Absolutely! But don't just copy — understand why log₂(8) = 3. Hint: 2 × 2 × 2 = 8. Understanding beats memorizing, and your test scores will thank you.