A logarithm answers the question: 'To what power must the base be raised to get this number?' For example, log₁₀(100) = 2 because 10² = 100. The logarithm is the inverse operation of exponentiation.

What is the difference between ln, log, log₁₀, and log₂?

ln (natural log) uses base e ≈ 2.71828. log₁₀ (common log) uses base 10 and is often written as just 'log'. log₂ (binary log) uses base 2, commonly used in computer science. 'log' with a custom base can be any positive number except 1.

How do I calculate log with a different base?

Use the change of base formula: logₐ(x) = ln(x) / ln(a) = log₁₀(x) / log₁₀(a). This calculator automatically applies this formula for any base you choose.

Antilog is the inverse of logarithm. If log_b(x) = y, then antilog_b(y) = x = b^y. For example, antilog₁₀(2) = 10² = 100. It converts a logarithmic value back to the original number.

Why can't I calculate log of a negative number or zero?

In real numbers, logarithms are only defined for positive values. The log of zero approaches negative infinity, and logs of negative numbers require complex numbers. This calculator works with real numbers only.

What are common uses of logarithms?

Logarithms are used in: measuring earthquake intensity (Richter scale), sound levels (decibels), pH in chemistry, compound interest calculations, algorithm complexity in computer science (O(log n)), and data compression.

Logarithm Calculator

Logarithms convert multiplicative relationships into additive ones, making them essential in acoustics (decibels), seismology (Richter scale), chemistry (pH), information theory (bits and entropy), and financial modeling (continuously compounded interest). This calculator evaluates log in any base, natural log (ln), and log₂ — and supports the change-of-base formula for converting between bases. Whether you are solving exponential equations, analyzing signal strength, or studying algorithmic complexity (Big O notation), logarithms are the tool of choice.

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functions Logarithm Calculator

Calculation Type

Value (x)

Must be a positive number

Base (b)

Must be positive and not equal to 1

Decimal Places

Log Rules

log(a×b) = log(a) + log(b)

log(a/b) = log(a) - log(b)

log(aⁿ) = n × log(a)

calculate Result

Expression

log₁₀(100)

Result

Verification

10² = 100 ✓

Same Value in Other Bases

ln (base e) 4.605170

log₁₀ 2.000000

log₂ 6.643856

Change of Base Formula

log_b(x) = ln(x) / ln(b)

Input Values

Calculation Type

expand_more

Value (x)

Base (b)

Decimal Places

expand_more

lightbulb Tips

•log_b(x) = y means b^y = x
•ln uses base e ≈ 2.718
•log(ab) = log(a) + log(b)
•log(a^n) = n × log(a)

functions Reference

Common Logs

log₁₀(10) = 1log₁₀(100) = 2 log₂(8) = 3log₂(16) = 4 ln(e) = 1ln(1) = 0

Constants

e ≈ 2.71828182845904

How to Use This Calculator

calculate

Choose Type

Select log with base, natural log (ln), log₁₀, log₂, or antilog.

edit

Enter Value

Input the number to calculate the logarithm of.

tune

Set Base

For custom log, enter the base (default: 10).

visibility

View Results

See the result with verification and steps.

The Formula

A logarithm answers: 'To what power must the base be raised to get this number?' It's the inverse of exponentiation.

log_b(x) = y means b^y = x

lightbulb Variables Explained

log_b(x) Logarithm of x with base b
ln(x) Natural logarithm (base e)
log(x) Common logarithm (base 10)

tips_and_updates Pro Tips

log₁₀(x) is called 'common log' and often written as just 'log'

ln(x) is natural log with base e ≈ 2.71828

log₂(x) is binary log, commonly used in computer science

Logarithm is the inverse of exponentiation: if bˣ = y, then log_b(y) = x

log(a×b) = log(a) + log(b) - useful for simplifying calculations

Change of base: log_a(x) = ln(x)/ln(a) = log(x)/log(a)

Calculate logarithms with any base, natural log (ln), common log (log₁₀), binary log (log₂), and antilog. Get step-by-step solutions.

Understanding Logarithms

Logarithms are the inverse of exponentiation. If 10² = 100, then log₁₀(100) = 2. They're essential in science, engineering, and computing for handling very large or small numbers.

Types of Logarithms

Common log (log₁₀) uses base 10. Natural log (ln) uses base e ≈ 2.718. Binary log (log₂) uses base 2. Each has specific applications in different fields.

Frequently Asked Questions

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