A logarithm answers the question: to what power must the base be raised to produce a given number? log_b(x) = y means b^y = x.

Can you take the log of 0 or a negative number?

No. log(0) is undefined (approaches −∞). The log of a negative number is undefined in the real number system (it requires complex numbers).

Logarithm Calculator — log₁₀, ln, log₂ & Custom Base

Input

Value (x)

Base

Base b =

Result

log₁₀(100) = 2

log₁₀(100)

ln(x)

4.60517019

log₂(x)

6.64385619

Result

Inverse bˆresult

100

Change of Base

Logarithm Curve

Base b =

Product Rule

log_b(M×N) = log_b(M) + log_b(N)

Quotient Rule

log_b(M/N) = log_b(M) − log_b(N)

Power Rule

log_b(Mⁿ) = n × log_b(M)

n (exponent)

Change of Base

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

Value (x)

From base

To base

Common Log Values

x	log₁₀(x)	ln(x)	log₂(x)
1	0	0	0
2	0.30103	0.693147	1
e ≈ 2.718	0.434294	1	1.442695
5	0.69897	1.609438	2.321928
10	1	2.302585	3.321928
20	1.30103	2.995732	4.321928
50	1.69897	3.912023	5.643856
π ≈ 3.1416	0.497150	1.144729	1.651496
100	2	4.605170	6.643856
1,000	3	6.907755	9.965784
e² ≈ 7.389	0.868589	2	2.885390

Log Law Summary

Law	Formula
Product	log_b(MN) = log_bM + log_bN
Quotient	log_b(M/N) = log_bM − log_bN
Power	log_b(Mⁿ) = n ⋅ log_bM
Change	log_b(x) = ln(x) / ln(b)
Identity	log_b(b) = 1
Zero	log_b(1) = 0
Inverse	b^log_b(x) = x

How to Use This Calculator

Enter a value

Type a positive number in the Value (x) field. The calculator updates instantly as you type.

Choose a base

Select log₁₀ (common), ln (natural), log₂ (binary), or Custom to enter any base you need.

Read the results

The live equation, hero value, and stat grid show the result along with ln(x) and log₂(x) for comparison.

Explore & practice

Switch to the Log Laws tab to practice product, quotient, power, and change-of-base rules interactively.

Key Terms

Logarithm

The exponent to which a base must be raised to produce a given number. log_b(x) = y means b^y = x.

Base

The number that is repeatedly multiplied. Must be positive and not equal to 1. Common bases: 10, e, 2.

Argument

The value inside the log, i.e. x in log_b(x). Must be positive for real-valued results.

Natural Log (ln)

Logarithm with base e ≈ 2.71828. Used extensively in calculus, physics, and finance.

Common Log

Logarithm with base 10. Written as log or log₁₀. Used for pH, decibels, and the Richter scale.

Antilog

The inverse of a logarithm: antilog_b(y) = b^y. If log₁₀(x) = 2, then antilog = 10² = 100.

Worked Examples

Example 1 — Common Log

Find log₁₀(1000)

Ask: 10 to what power = 1000?

10³ = 1000

log₁₀(1000) = 3

Example 2 — Natural Log

Find ln(e⁵)

Using the power rule: ln(e⁵) = 5 × ln(e)

ln(e) = 1, so 5 × 1 = 5

ln(e⁵) = 5

Example 3 — Custom Base

Find log₃(81)

3 to what power = 81?

3⁴ = 81

log₃(81) = 4

Example 4 — Change of Base

Find log₅(200) using base 10

= log₁₀(200) / log₁₀(5)

= 2.30103 / 0.69897

log₅(200) ≈ 3.292

Logarithms in the Real World

Sound and Decibels

The human ear can detect an enormous range of sound intensities — from a barely perceptible whisper to a jet engine. Because this range spans 12 orders of magnitude, a linear scale would be impractical. Decibels use a base-10 logarithm: dB = 10 × log₁₀(I / I₀), where I₀ = 10⁻¹² W/m². A normal conversation (60 dB) is 10&sup6; times more intense than the threshold of hearing.

pH and Chemistry

The pH scale measures acidity: pH = −log₁₀[H⊃]. Pure water at 25°C has [H⊃] = 10⁻⁷ mol/L, giving pH 7. Each pH unit represents a 10× change in concentration — vinegar at pH 3 is 10,000 times more acidic than water.

Earthquake Magnitude

The Richter scale uses M = log₁₀(A/A₀). A magnitude 7 earthquake releases roughly 32 times more energy than a magnitude 6, because each whole number increase corresponds to about 31.6× more seismic energy.

Information Theory and Computers

Binary logarithms (log₂) underpin computer science. The number of bits needed to represent n possibilities is ⌈log₂(n)⌉. This is why 8 bits can store 2⁸ = 256 values, and why searching a sorted list of n items takes O(log n) comparisons.

Finance and Doubling Time

Compound growth uses natural logarithm to solve for time: t = ln(FV/PV) / ln(1 + r). The Rule of 72 estimates doubling time as 72/r%, derived from the logarithm approximation ln(2) / r ≈ 0.693/r.

Frequently Asked Questions

What is the difference between log and ln?

log (or log₁₀) is the common logarithm with base 10. ln is the natural logarithm with base e ≈ 2.71828. They are both logarithms — just with different bases. ln is preferred in calculus and physics; log₁₀ is used in chemistry (pH) and acoustics (dB).

Can you take the logarithm of 0 or a negative number?

No — in the real number system. log(0) approaches −∞ (undefined). The logarithm of a negative number requires complex numbers. This calculator shows an error for x ≤ 0.

What does the inverse / antilog mean?

The antilog is the inverse operation: if log_b(x) = y, then bˆy = x. For example, log₁₀(100) = 2, so antilog₁₀(2) = 10² = 100. This calculator shows the inverse as a verification step.

What is the change-of-base formula and when do I need it?

log_b(x) = ln(x) / ln(b) = log₁₀(x) / log₁₀(b). You need it when your calculator only has ln or log₁₀ buttons but you want a different base — or when simplifying expressions with mixed bases.

Why is log₂ important in computing?

Binary logarithms directly measure the number of bits in a value. log₂(n) tells you how many times you can halve n before reaching 1 — which is exactly the number of steps in binary search, and the depth of a balanced binary tree.

What are logarithm graphs shaped like?

Logarithm curves are concave, slowly increasing functions. They pass through (1, 0) for any base — because log_b(1) = 0. They approach −∞ as x → 0⁺, and grow without bound but very slowly. Higher bases result in flatter curves; bases less than 1 produce decreasing curves.

Logarithm Calculator