# Log vs. Ln — What's the Difference?

By Tayyaba Rehman — Published on January 13, 2024
Log is the general logarithm, often base 10 or base e (Ln), while Ln is specifically the natural logarithm with base e (approximately 2.71828).

## Key Differences

Logarithms are exponents to which a fixed number, the base, must be raised to produce a given number. "Log" commonly represents a logarithm with a base of 10 in general mathematics. For example, log(100) equals 2 because 10 squared is 100.
The notation "Ln" stands for the natural logarithm, which specifically has a base of e, where e is an irrational and transcendental number approximately equal to 2.71828. Ln is prevalent in higher mathematics, particularly in calculus, due to its unique properties when differentiating and integrating.
In practical applications, log is used in many scientific calculations, such as pH in chemistry or the Richter scale in geology, which are based on logarithms of base 10. This makes log an essential concept in a wide range of scientific and engineering disciplines.
Conversely, Ln is critical in pure mathematics, especially where growth processes are analyzed, such as in the case of compound interest in finance or population growth in biology. This makes the natural logarithm indispensable in both theoretical and applied mathematics.
Calculators and computers typically have specific keys or functions for both log and ln, demonstrating their distinct uses. While log is often used in contexts where ratios and orders of magnitude are important, ln is usually applied in continuous growth contexts and in solving time-related problems where e is the base of the exponential function.

## Comparison Chart

### Base

Commonly 10 or unspecified
Always e (≈2.71828)

Log(x)
Ln(x)

### Origin of Term

From "logarithm"
Abbreviation for "logarithmus naturalis"

### Use in Calculators

Usually labeled as "LOG"
Usually labeled as "LN"

### Field of Use

General mathematics, pH levels, decibels
Higher mathematics, calculus, natural processes

## Compare with Definitions

#### Log

A detailed record of events
The captain kept a log of the voyage.

#### Ln

A function in calculus for differentiation
The derivative of Ln(x) is 1/x.

#### Log

A unit of wood cut for fuel
He added a log to the fire.

#### Ln

The natural logarithm function
The value of Ln(1) is always 0.

#### Log

To enter information in a record
She logged her research findings.

#### Ln

Integral in solving exponential equations
Ln is used to solve for time in physics.

#### Log

A usually large section of a trunk or limb of a fallen or felled tree.

#### Ln

Mathematical concept for growth analysis
Ln helps understand compound interest.

#### Log

A long thick section of trimmed, unhewn timber.

#### Ln

A key on scientific calculators
Press Ln to calculate natural logarithms.

#### Log

A device trailed from a ship to determine its speed through the water.

#### Ln

Abbreviation of lane

#### Log

A record of a ship's speed, its progress, and any shipboard events of navigational importance.

#### Log

The book in which this record is kept.

#### Log

A record of a vehicle's performance, as the flight record of an aircraft.

#### Log

A record, as of the performance of a machine or the progress of an undertaking
A computer log.
A trip log.

A logarithm.

#### Log

To cut down, trim, and haul the timber of (a piece of land).

#### Log

To cut (timber) into unhewn sections.

#### Log

To enter in a record, as of a ship or an aircraft.

#### Log

To travel (a specified distance, time, or speed)
Logged 30,000 air miles in April.

#### Log

To spend or accumulate (time)
Had logged 25 years with the company.

#### Log

To cut down, trim, and haul timber.

#### Log

The trunk of a dead tree, cleared of branches.
They walked across the stream on a fallen log.

#### Log

Any bulky piece as cut from the above, used as timber, fuel etc.

#### Log

Anything shaped like a log; a cylinder.

#### Log

(nautical) A floating device, usually of wood, used in navigation to estimate the speed of a vessel through water.

#### Log

(figuratively) A blockhead; a very stupid person.

#### Log

A heavy longboard.

#### Log

(figuratively) A rolled cake with filling.

#### Log

(mining) A weight or block near the free end of a hoisting rope to prevent it from being drawn through the sheave.

#### Log

(vulgar) A piece of feces.

#### Log

(vulgar) A penis.

#### Log

A logbook, or journal of a vessel's (or aircraft's) progress.

#### Log

A chronological record of actions, performances, computer/network usage, etc.

#### Log

(computer science) Specifically, an append-only sequence of records written to file.

#### Log

A Hebrew unit of liquid volume (about 3{{nbsp}}liter).

#### Log

Synonym of logarithm. Category:en:Functions
To multiply two numbers, add their logs.

#### Log

(sciences) A difference of one in the logarithm, usually in base 10; an order of magnitude.

#### Log

(transitive) To cut trees into logs.

#### Log

(transitive) To cut down (trees).

#### Log

(intransitive) To cut down trees in an area, harvesting and transporting the logs as wood.

#### Log

(transitive) To make, to add an entry (or more) in a log or logbook.
To log the miles travelled by a ship

#### Log

(transitive) To travel (a distance) as shown in a logbook.

#### Log

(transitive) To travel at a specified speed, as ascertained by a chip log.

#### Log

A Hebrew measure of liquids, containing 2.37 gills.

#### Log

A bulky piece of wood which has not been shaped by hewing or sawing.

#### Log

An apparatus for measuring the rate of a ship's motion through the water.

#### Log

The record of the rate of speed of a ship or airplane, and of the course of its progress for the duration of a voyage; also, the full nautical record of a ship's cruise or voyage; a log slate; a log book.

#### Log

A record and tabulated statement of the person(s) operating, operations performed, resources consumed, and the work done by any machine, device, or system.

#### Log

A weight or block near the free end of a hoisting rope to prevent it from being drawn through the sheave.

#### Log

A record of activities performed within a program, or changes in a database or file on a computer, and typically kept as a file in the computer.

#### Log

To enter in a ship's log book; as, to log the miles run.

#### Log

To record any event in a logbook, especially an event relating to the operation of a machine or device.

#### Log

To engage in the business of cutting or transporting logs for timber; to get out logs.

#### Log

To move to and fro; to rock.

#### Log

A segment of the trunk of a tree when stripped of branches

#### Log

Large log at the back of a hearth fire

#### Log

The exponent required to produce a given number

#### Log

A written record of messages sent or received;
They kept a log of all transmission by the radio station
An email log

#### Log

A written record of events on a voyage (of a ship or plane)

#### Log

Measuring instrument that consists of a float that trails from a ship by a knotted line in order to measure the ship's speed through the water

#### Log

Enter into a log, as on ships and planes

#### Log

Cut lumber, as in woods and forests

#### Log

A part of a tree trunk
They used a log to create a makeshift bridge.

#### Log

To travel a certain distance or time
The pilot logged 300 hours of flight time.

## Common Curiosities

#### What does "Ln" represent?

"Ln" represents the natural logarithm with base e.

#### Why is Ln important in calculus?

Ln has properties that simplify the differentiation and integration of exponential functions.

#### Is there a relationship between Log and Ln?

Yes, they are related through the change of base formula.

#### What does "Log" stand for?

"Log" stands for logarithm, usually base 10 or unspecified.

Use Log for base 10 logarithms or when the base is not specified, common in general mathematics and practical applications.

#### Can Log have bases other than 10?

Yes, Log can have any positive base, though base 10 is most common.

#### Why is e the base for Ln?

The number e (approx. 2.71828) has properties that make it the natural choice for continuous growth processes.

#### How do you calculate Log and Ln without a calculator?

You can use series expansions or refer to logarithm tables for approximate values.

#### What is the base of a common Log?

The base of a common Log is typically 10.

#### Are Log and Ln interchangeable?

No, they are used for different bases and are not interchangeable.

#### Are Log and Ln used in computer programming?

Yes, programming languages often include functions for both Log and Ln.

#### Is Ln just a type of Log?

Yes, Ln is a specific type of Log with base e.

#### What does Log(0) or Ln(0) equal?

Log(0) and Ln(0) are undefined because no number raised to a power will result in zero.

#### Can you convert between Log and Ln?

Yes, using the formula Log(x) = Ln(x) / Ln(10).

#### What is the inverse function of Ln?

The inverse function of Ln is the exponential function e^x.