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Log Calculator,
ln, log₁₀, log₂, any base.

Calculate the natural logarithm (ln), common log (log₁₀), binary log (log₂), and any arbitrary-base logarithm simultaneously from a single input. Includes a step-by-step change-of-base derivation, interactive curve chart, and a logarithm identities reference.

How it works4 bases at once

Inputs

Logarithm

Computing

log_10(8) = 0.90308999

Number (x)

Base

Quick values

ln(x): 2.0794415
log₁₀(x): 0.90308999
log₂(x): 3

Logarithms of 8

x = 8

Natural log

ln(8)

2.0794415

base e ≈ 2.71828

Common log

log₁₀(8)

0.90308999

base 10

Binary log

log₂(8)

base 2

Custom base

logᵦ(x)

← set base b

switch mode to 'b'

Math notepad

Step-by-step using the change-of-base identity.

handworked

Natural logarithm (base e ≈ 2.71828...):

ln(8) = 2.0794415

Common logarithm (base 10):

log₁₀(8) = ln(8) / ln(10)

= 2.0794415 / 2.3025851

= 0.90308999

Binary logarithm (base 2) via change-of-base:

log₂(8) = ln(8) / ln(2)

= 2.0794415 / 0.69314718

= 3

Logarithm curves

y = log(x) for three common bases

lnlog₁₀log₂

Reference

Logarithm identities

Product rulelog_b(x·y) = log_b(x) + log_b(y)

Quotient rulelog_b(x/y) = log_b(x) − log_b(y)

Power rulelog_b(xⁿ) = n · log_b(x)

Change of baselog_b(x) = ln(x) / ln(b)

Reciprocallog_b(1/x) = −log_b(x)

log of 1log_b(1) = 0 (for any b)

log of baselog_b(b) = 1 (for any b)

Logarithm guide

What is a logarithm and how do you calculate one?

A logarithm answers the question: to what power must a given base be raised to produce a given number? If b^y = x, then by definition log_b(x) = y. Logarithms are the inverse operation of exponentiation, just as subtraction undoes addition, the logarithm undoes raising to a power.

The formal definition

log_b(x) = y ⟺ b^y = x (where b > 0, b ≠ 1, and x > 0)

For example, log₁₀(1000) = 3 because 10³ = 1000; log₂(8) = 3 because 2³ = 8; and ln(e) = 1 because e¹ = e.

Natural logarithm: ln(x)

The natural logarithm uses Euler's number e ≈ 2.71828 as its base. It appears throughout calculus, physics, finance, and statistics because the derivative of ln(x) is simply 1/x — cleaner than any other logarithmic base:

d/dx [ln(x)] = 1/x

Natural log underpins compound growth models, probability distributions (including the normal distribution's Z-score), and information theory. When mathematicians write "log" without a subscript, they often mean ln.

Common logarithm — log₁₀(x)

The common logarithm (base 10) was the workhorse of pre-computer arithmetic (tables of log₁₀ values let engineers and scientists multiply large numbers by adding their logs. It remains useful wherever the decimal system is natural:

The pH scale: pH = −log₁₀([H⁺])
The Richter earthquake scale: magnitude = log₁₀(amplitude)
Decibels: dB = 10 × log₁₀(P₂/P₁)
The magnitude of a star in astronomy

The number of decimal digits in a positive integer n equals ⌊log₁₀(n)⌋ + 1, a handy shortcut in programming.

Binary logarithm) log₂(x)

The binary logarithm is the natural unit for information theory and computer science, as defined by Claude Shannon in 1948. One bit of information equals log₂(2) = 1. Key uses:

Big-O complexity: A binary search over n items takes O(log₂ n) comparisons.
Data compression: The optimal number of bits to encode an event with probability p is −log₂(p) bits.
Number of bits: ⌊log₂(n)⌋ + 1 is the number of bits needed to represent the integer n in binary.

The change-of-base formula

Any logarithm can be expressed in terms of any other base using the change-of-base identity. This is how the calculator computes arbitrary-base logarithms from the built-inMath.log (which computes ln):

log_b(x) = ln(x) / ln(b) = log₁₀(x) / log₁₀(b) = log₂(x) / log₂(b)

For example, log₅(125) = ln(125) / ln(5) = 4.8283 / 1.6094 = 3. You can verify: 5³ = 125 ✓.

Essential logarithm properties

These six identities let you simplify virtually any logarithmic expression:

Rule	Formula	Example
Product	log_b(xy) = log_b(x) + log_b(y)	log₂(4·8) = log₂(4) + log₂(8) = 2 + 3 = 5
Quotient	log_b(x/y) = log_b(x) − log_b(y)	log₁₀(100/10) = 2 − 1 = 1
Power	log_b(xⁿ) = n · log_b(x)	ln(e⁵) = 5 · ln(e) = 5
Root	log_b(ⁿ√x) = log_b(x) / n	log₁₀(√100) = log₁₀(100) / 2 = 1
log of 1	log_b(1) = 0	ln(1) = 0, log₂(1) = 0
log of base	log_b(b) = 1	ln(e) = 1, log₁₀(10) = 1

Domain, range, and special values

Logarithms are only defined for positive real numbers:

log(0) is undefined. As x approaches 0 from the positive side, log(x) → −∞. This is not a real number.
log(negative) is not a real number (it is complex: ln(−x) = ln(x) + πi, but that requires complex arithmetic).
log(1) = 0 for any base, because b⁰ = 1.
log(x) < 0 when 0 < x < 1 — logarithms of fractions are negative.

x	ln(x)	log₁₀(x)	log₂(x)
0.001	−6.9078	−3	−9.9658
0.1	−2.3026	−1	−3.3219
0.5	−0.6931	−0.3010	−1
1	0	0	0
2	0.6931	0.3010	1
e	1	0.4343	1.4427
10	2.3026	1	3.3219
100	4.6052	2	6.6439
1000	6.9078	3	9.9658

Where logarithms appear in real life

Logarithms compress wide-ranging data into manageable scales and appear wherever growth, decay, or sensitivity follows a multiplicative pattern:

Finance: The Black-Scholes option pricing formula uses ln(S/K) to model stock price ratios. Continuous compounding uses e^(r·t). Annualised returns compound multiplicatively, so log-returns add, making statistics tractable.
Science: pH, sound decibels, earthquake magnitude, star brightness (apparent magnitude) and stellar luminosity all use log-scaled measurements.
Statistics: Log-normal distributions model stock prices, city sizes, and income distributions. Log transformation normalises right-skewed data.
Computer science: Time complexity of divide-and-conquer algorithms (merge sort = O(n log₂ n)), binary search (O(log₂ n)), and Shannon entropy all use log₂.

FAQ

Frequently asked

What is the difference between log and ln?+

'log' can mean different things in different contexts. In mathematics and engineering, 'log' usually means log₁₀ (common logarithm). In pure mathematics and physics, 'log' often means ln (natural logarithm). This calculator always shows both explicitly, log₁₀(x) and ln(x), to avoid ambiguity.

Why is log(0) undefined?+

Because there is no finite power to which any positive base b can be raised to equal zero. As x approaches 0 from the right, log_b(x) approaches −∞ (negative infinity). Zero is not in the domain of any logarithm.

What does log₂(x) = n mean in computer science?+

It means you need to divide x in half n times to get to 1. This is the number of steps in a binary search over x items. For example, log₂(1,000,000) ≈ 19.93, meaning binary search over a million sorted items takes at most 20 comparisons.

How do I calculate log of a negative number?+

Real-valued logarithms are not defined for negative numbers. The complex extension is: ln(−x) = ln(x) + πi, where i = √(−1). This gives a complex result. This calculator only supports positive real inputs.

What is the change-of-base formula used for?+

It lets you compute a logarithm in any base using a calculator or programming language that only provides ln or log₁₀. The formula is: log_b(x) = ln(x) / ln(b). All logarithms in this calculator are computed this way internally, only ln is a primitive.

How do you undo a logarithm (antilog)?+

The inverse of log_b(x) is b^x. To undo ln(x), compute e^x. To undo log₁₀(x), compute 10^x. To undo log₂(x), compute 2^x. These are called antilogarithms or simply exponentials.

What is log₁₀(0.5)?+

log₁₀(0.5) = log₁₀(1/2) = −log₁₀(2) ≈ −0.30103. This is negative because 0.5 < 1, and logarithms of numbers less than 1 are always negative (for any base b > 1). Verify: 10^(−0.30103) ≈ 0.5 ✓.

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