Natural Logarithm — Definition, Formula & Examples

Q: What is the difference between ln and log?

The notation ln always means the logarithm base $e$ (the natural logarithm). The notation log, without a subscript, usually means the common logarithm (base 10) in applied sciences and on most calculators, but in higher mathematics it often means base $e$. To avoid confusion, check the context or use the explicit notations $\ln$ for base $e$ and $\log_{10}$ for base 10.

Q: Why can't you take the natural logarithm of a negative number or zero?

Because $e^y$ is always positive for every real number $y$, there is no real exponent that makes $e^y$ equal to zero or a negative number. Therefore $\ln x$ is defined only for $x > 0$ when working with real numbers.

Natural Logarithm

The logarithm base e of a number. That is, the power of e necessary to equal a given number. The natural logarithm of x is written ln x. For example, ln 8 is 2.0794415... since e^2.0794415... = 8.

See also

Common logarithm, logarithm rules

Key Formula

\ln x = y \quad \Longleftrightarrow \quad e^y = x

Where:

$x$ = The positive real number you are taking the logarithm of (the argument)
$y$ = The exponent to which e must be raised to produce x
$e$ = Euler's number, approximately 2.71828

Worked Example

Problem: Find the natural logarithm of 1, e, and e³.

Step 1: Find ln 1. Ask: what power of e gives 1? Since any number raised to the power 0 equals 1:

\ln 1 = 0 \quad \text{because} \quad e^0 = 1

Step 2: Find ln e. Ask: what power of e gives e? Raising e to the first power gives e:

\ln e = 1 \quad \text{because} \quad e^1 = e

Step 3: Find ln(e³). Ask: what power of e gives e³? The answer is simply the exponent itself:

\ln(e^3) = 3 \quad \text{because} \quad e^3 = e^3

Answer: ln 1 = 0, ln e = 1, and ln(e³) = 3.

Another Example

Problem: Solve the equation ln(2x) = 5 for x.

Step 1: Convert the natural logarithm equation to exponential form. If ln(2x) = 5, then by definition:

2x = e^5

Step 2: Solve for x by dividing both sides by 2:

x = \frac{e^5}{2}

Step 3: Compute the approximate value using e⁵ ≈ 148.413:

x \approx \frac{148.413}{2} \approx 74.207

Answer: x = e⁵/2 ≈ 74.207

Frequently Asked Questions

What is the difference between ln and log?

The notation ln always means the logarithm base

e

(the natural logarithm). The notation log, without a subscript, usually means the common logarithm (base 10) in applied sciences and on most calculators, but in higher mathematics it often means base

e

. To avoid confusion, check the context or use the explicit notations

\ln

for base

e

and

\log_{10}

for base 10.

Why can't you take the natural logarithm of a negative number or zero?

Because

e^y

is always positive for every real number

y

, there is no real exponent that makes

e^y

equal to zero or a negative number. Therefore

\ln x

is defined only for

x > 0

when working with real numbers.

Natural Logarithm (ln) vs. Common Logarithm (log₁₀)

Both are logarithms, but they use different bases. The natural logarithm uses base

e \approx 2.71828

and is written

\ln x

. The common logarithm uses base 10 and is written

\log_{10} x

or simply

\log x

on calculators. They are related by the change-of-base formula:

\ln x = \frac{\log_{10} x}{\log_{10} e} \approx \frac{\log_{10} x}{0.43429}

. The natural logarithm appears most often in calculus and continuous growth models, while the common logarithm is convenient for orders of magnitude and scales like pH or decibels.

Why It Matters

The natural logarithm is central to calculus because

\frac{d}{dx}\ln x = \frac{1}{x}

, one of the simplest and most important derivative formulas. It models continuous growth and decay—compound interest, radioactive decay, and population growth all use

\ln

in their equations. Whenever you need to "undo" an exponential involving

e

, the natural logarithm is the tool you reach for.

Common Mistakes

Mistake: Thinking that ln(a + b) equals ln a + ln b.

Correction: The logarithm sum rule says ln a + ln b = ln(ab), not ln(a + b). There is no simple formula for the natural log of a sum.

Mistake: Confusing ln x with 1/x or treating ln as a multiplier.

Correction: ln is a function, not a coefficient. Writing "ln · x" or canceling ln from both sides of ln(x) = ln(y) + 3 as if ln were a factor is incorrect. Instead, use the definition or logarithm rules to manipulate such equations properly.

Related Terms

Logarithm — General concept; ln is a specific case
e — The base of the natural logarithm
Common Logarithm — Logarithm base 10, often compared to ln
Base of a Logarithm — Determines which logarithm you are using
Logarithm Rules — Product, quotient, and power rules apply to ln
Power — ln reverses exponentiation with base e