What is the common logarithm?

The common logarithm is log base 10, written as log(x) without a subscript. It measures how many powers of 10 a number contains. For example, log(100) = 2 because 10² = 100.

What is the natural logarithm?

The natural logarithm is log base e (≈ 2.71828), written as ln(x). It appears naturally in calculus and continuous growth/decay problems. The derivative of ln(x) is 1/x.

What is the number e?

The number e ≈ 2.71828 is an irrational constant that emerges from continuous compound interest: e = lim(1 + 1/n)^n as n→∞. It's the base of natural logarithms and appears throughout mathematics.

What is the difference between log and ln?

Log typically means base 10 (common logarithm), while ln means base e (natural logarithm). On calculators, LOG is base 10 and LN is base e. Some programming languages use log for natural log — always check conventions.

When should I use log vs ln?

Use log (base 10) for orders of magnitude, decibels, pH, and Richter scales. Use ln (base e) for calculus, continuous growth/decay, and theoretical work. For solving equations, either works with change of base.

How do you convert between log and ln?

Use ln(x) = log(x) × ln(10) ≈ 2.303 × log(x), or log(x) = ln(x) / ln(10) ≈ ln(x) / 2.303. This follows from the change of base formula.

Why is e important in mathematics?

The function e^x is its own derivative — no other base has this property. This makes e natural for calculus, differential equations, compound interest, and exponential growth/decay modeling.

What is ln(1) and ln(e)?

ln(1) = 0 because e⁰ = 1. ln(e) = 1 because e¹ = e. These are the two fundamental reference points for the natural logarithm function.

How do ln(x) and log(x) graphs compare?

Both pass through (1, 0) and have vertical asymptotes at x = 0. Since e log(x) for all x > 1. They differ by a constant factor: ln(x) ≈ 2.303 × log(x).

How do you compute log base 2 on a calculator?

Use the change of base formula: log₂(x) = log(x)/log(2) or ln(x)/ln(2). For example, log₂(8) = log(8)/log(2) = 0.903/0.301 = 3.

Common Natural Logarithms

Key Terms

The Common Logarithm

The Natural Logarithm

The Number e

Calculator Conventions

When to Use Which

Converting Between Bases

Comparing Graphs

Side-by-Side Summary

Two Bases That Dominate

Among infinitely many possible bases, two have earned special notation and widespread use. The common logarithm uses base

10

, aligning with the decimal system and appearing throughout applied sciences. The natural logarithm uses base

e \approx 2.71828

, emerging from calculus and dominating theoretical mathematics. Understanding when to use each — and how to convert between them — is essential for practical computation.

Key Terms

The Two Special Bases

Common Logarithm— base

10

, written

\log(x)

Natural Logarithm— base

e

, written

\ln(x)

Euler's Number (e)— the irrational constant

e \approx 2.71828

, base of the natural logarithm

Converting

Change of Base Formula— converts between any two bases using

\log

\ln

See All Algebra Definitions →

The Common Logarithm

The common logarithm is the logarithm with base

10

. It is written

\log(x)

without a subscript, or

\log_{10}(x)

when clarity is needed.

Base

10

connects directly to the decimal number system. Each power of

10

corresponds to a place value:

10^1 = 10

10^2 = 100

10^3 = 1000

. The common logarithm inverts this relationship, reporting how many powers of ten a number contains.

For powers of

10

, the values are immediate:

\log(10) = 1

\log(100) = 2

\log(1000) = 3

\log(0.1) = -1

\log(0.01) = -2

. For other numbers, the integer part indicates the order of magnitude. Since

\log(500) \approx 2.699

, the number

500

lies between

10^2 = 100

and

10^3 = 1000

, closer to the latter.

Historically, common logarithms enabled calculation before electronic computers. Logarithm tables converted multiplication into addition — a far simpler operation to perform by hand. Though tables are obsolete, the notation persists, and base

10

remains natural for human-scale quantities.

The Natural Logarithm

The natural logarithm is the logarithm with base

e

, where

e \approx 2.71828

. It is written

\ln(x)

, read "natural log of

x

," or occasionally

\log_e(x)

.

The number

e

is not chosen arbitrarily. It emerges from the study of continuous growth: if a quantity grows at a rate proportional to its current size,

e

appears in the formula describing that growth. Compound interest computed with infinitely many compounding periods, radioactive decay, population dynamics — all involve

e

naturally.

In calculus,

e

has a unique property: the derivative of

e^x

e^x

itself. No other base produces this self-replicating behavior under differentiation. The natural logarithm inherits a corresponding property: the derivative of

\ln(x)

1/x

, the simplest possible form.

The values

\ln(1) = 0

and

\ln(e) = 1

follow from the general properties. Additional reference points:

\ln(2) \approx 0.693

\ln(10) \approx 2.303

. These approximations appear frequently in applications.

The Number e

The constant

e

is an irrational number approximately equal to

2.71828

. Its decimal expansion continues without repeating:

e = 2.718281828459045...

One definition comes from compound interest. If

\

earns

100\%

interest per year, compounded

times, the year-end balance is

(1 + 1/n)^n

. As

increases — daily, hourly, every second, continuously — this expression approaches

e$:

e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n

Another definition uses an infinite series:

e = 1 + 1 + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots = \sum_{n=0}^{\infty} \frac{1}{n!}

The number appears throughout mathematics far beyond logarithms — in probability, complex analysis, differential equations, and number theory. Its ubiquity justifies elevating

\log_e

to the status of "natural."

Three independent definitions of

e

are now in play — the compound-interest limit, the infinite series, and the calculus characterization mentioned in the previous section. The table below puts them side by side; each produces the same irrational constant from a different starting point.

Definition	Formula	Where it comes from
Compound interest limit	e = lim (1 + 1⁄n)ⁿ as n → ∞	100% interest on $1 with infinitely many compounding periods
Infinite series	e = Σ 1⁄n! = 1 + 1 + 1⁄2! + 1⁄3! + ...	Taylor series of eˣ evaluated at x = 1
Calculus characterization	unique base a for which d⁄dx aˣ = aˣ	the exponential function that is its own derivative

Calculator Conventions

Scientific calculators and programming languages use varying conventions for logarithm notation, which can cause confusion.

Most scientific calculators have two buttons:

\log

for the common logarithm (base

10

) and

\ln

for the natural logarithm (base

e

). This matches standard mathematical notation. To compute a logarithm with a different base, the change of base formula is required.

Some programming languages diverge. In Python, the

\texttt{math.log(x)}

function computes the natural logarithm by default, not the common logarithm. The common logarithm requires

\texttt{math.log10(x)}

. Other languages follow similar patterns — always check documentation.

Spreadsheet software like Excel uses

\texttt{LOG(x)}

for common logarithm and

\texttt{LN(x)}

for natural logarithm, matching calculator conventions. The function

\texttt{LOG(x, base)}

allows arbitrary bases.

When in doubt, test with known values. If the software returns

1

for

\log(10)

, it uses base

10

. If it returns

1

for

\log(e)

, it uses base

e

Tool	Common log (base 10)	Natural log (base e)	Note
Scientific calculator	LOG key	LN key	matches standard mathematical notation
Python (math module)	math.log10(x)	math.log(x)	trap: math.log defaults to natural log, not common
Excel / Google Sheets	LOG(x) or LOG10(x)	LN(x)	LOG(x, base) handles arbitrary bases

When to Use Which

The choice between common and natural logarithms depends on context.

Use common logarithms when working with orders of magnitude, human-readable scales, or decimal-based measurements. The Richter scale for earthquakes, the decibel scale for sound intensity, and the pH scale for acidity all employ base

10

. A change of

1

on these scales represents a factor of

10

— intuitive for decimal-system thinkers.

Use natural logarithms when working with continuous growth or decay, calculus, or theoretical derivations. Exponential models of the form

Ae^{kt}

pair naturally with

\ln

. Solving

e^{2x} = 5

is cleanest with natural logarithms:

2x = \ln(5)

, so

x = \ln(5)/2

.

For solving general exponential equations, either logarithm works. The equation

3^x = 7

can be solved as

x = \log(7)/\log(3)

x = \ln(7)/\ln(3)

— both yield the same numerical answer. Choose whichever is available or conventional in the given context.

Converting Between Bases

The change of base formula from logarithm rules enables conversion between common and natural logarithms:

\log_a(x) = \frac{\log_b(x)}{\log_b(a)}

To convert a logarithm of any base to common logarithms:

\log_a(x) = \frac{\log(x)}{\log(a)}

To convert to natural logarithms:

\log_a(x) = \frac{\ln(x)}{\ln(a)}

For example,

\log_2(10) = \frac{\ln(10)}{\ln(2)} = \frac{2.303}{0.693} \approx 3.322

. Verification:

2^{3.322} \approx 10

.

Converting between common and natural logarithms directly uses

\ln(10) \approx 2.303

\ln(x) = \log(x) \cdot \ln(10) \approx 2.303 \cdot \log(x)

\log(x) = \frac{\ln(x)}{\ln(10)} \approx \frac{\ln(x)}{2.303}

These conversions allow computation with any base using only the

\log

and

\ln

buttons available on standard calculators.

Comparing Graphs

The graphs of

y = \log(x)

and

y = \ln(x)

share the same basic shape but differ in steepness.

Both pass through

(1, 0)

— all logarithms satisfy

\log_a(1) = 0

. The common logarithm passes through

(10, 1)

; the natural logarithm passes through

(e, 1) \approx (2.718, 1)

. Both have vertical asymptotes at

x = 0

and extend to

+\infty

x

increases.

Since

e < 10

, the natural logarithm reaches

1

at a smaller input value. This makes

\ln(x)

grow faster initially and appear steeper for small

x

. For any positive

x > 1

\ln(x) > \log(x)

because the natural logarithm uses a smaller base.

The ratio between them is constant:

\ln(x) = \ln(10) \cdot \log(x) \approx 2.303 \cdot \log(x)

. The graphs are vertical stretches of each other. Multiplying every output of the common logarithm by

\ln(10)

produces the natural logarithm.

Side-by-Side Summary

Across base, key values, graph behavior, calculus role, applications, and conversion formulas, common and natural logarithms are two distinct tools that serve different purposes — but they live on the same family of curves and convert into each other through one constant. The table below pairs every aspect covered above so the two can be compared at a glance.

Aspect	Common log: log(x)	Natural log: ln(x)
Base	10	e ≈ 2.71828
Anchor (a, 1) on graph	(10, 1)	(e, 1) ≈ (2.718, 1)
log(2) reference value	log(2) ≈ 0.301	ln(2) ≈ 0.693
Steepness (for x > 1)	gentler — base 10 is larger	steeper — smaller base means faster initial rise
Calculus property	d⁄dx log(x) = 1 ⁄ (x · ln(10))	d⁄dx ln(x) = 1⁄x — cleanest possible
Where it shows up	Richter scale, decibels, pH, orders of magnitude	calculus, continuous growth/decay, theoretical derivations
Conversion to the other	log(x) = ln(x) ⁄ ln(10) ≈ ln(x) ⁄ 2.303	ln(x) = log(x) · ln(10) ≈ 2.303 · log(x)