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Common Natural Logarithms



Key Terms
The Common Logarithm
Logarithm Notation
The Natural Logarithm
The Number e
Calculator Conventions
When to Use Which
Converting Between Bases
Comparing Graphs
Side-by-Side Summary
Common and Natural Logarithms FAQ



Two Bases That Dominate

Among infinitely many possible bases, two have earned special notation and widespread use. The common logarithm uses base 1010, aligning with the decimal system and appearing throughout applied sciences. The natural logarithm uses base e≈2.71828e \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 1010, written log⁡(x)\log(x)
Natural Logarithm— base ee, written ln⁡(x)\ln(x)
Euler's Number (e)— the irrational constant e≈2.71828e \approx 2.71828, base of the natural logarithm

Converting

Change of Base Formula— converts between any two bases using log⁡\log or ln⁡\ln

See All Algebra Definitions →


The Common Logarithm

The common logarithm is the logarithm with base 1010. It is written log⁡(x)\log(x) without a subscript, or log⁡10(x)\log_{10}(x) when clarity is needed.

Base 1010 connects directly to the decimal number system. Each power of 1010 corresponds to a place value: 101=1010^1 = 10, 102=10010^2 = 100, 103=100010^3 = 1000. The common logarithm inverts this relationship, reporting how many powers of ten a number contains.

For powers of 1010, the values are immediate: log⁡(10)=1\log(10) = 1, log⁡(100)=2\log(100) = 2, log⁡(1000)=3\log(1000) = 3, log⁡(0.1)=−1\log(0.1) = -1, log⁡(0.01)=−2\log(0.01) = -2. For other numbers, the integer part indicates the order of magnitude. Since log⁡(500)≈2.699\log(500) \approx 2.699, the number 500500 lies between 102=10010^2 = 100 and 103=100010^3 = 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 1010 remains natural for human-scale quantities.

Logarithm Notation

Notation

Logarithm Notation

The subscript that names the base, the three-way war over what a bare log means, and the two Latin letters everyone reads differently.
ee and exp⁡\exp — exponential functions; the defining equivalence log⁡bx=y  ⟺  by=x\log_b x = y \iff b^y = x — the logarithms family page.
log⁡bx\log_b x
Log base b of x
Three slots: the base as a subscript, the argument on the line, and the output — which is an exponent: the power bb needs to reach xx. Like sin⁡x\sin x, the argument often goes unparenthesised: log⁡x\log x and log⁡(x)\log(x) are the same statement.
CasesSubscript present — unambiguous, any base. Subscript dropped — a dialect question, next entry. Two-argument software form — LOG(x, base), argument first, base second, as in Calculator Conventions below.
Also writtenlog⁡b(x)\log_b(x) with full parentheses, obligatory once the argument is a sum: log⁡b(x+1)\log_b(x+1), never log⁡bx+1\log_b x + 1.
Do not confuseA product. log⁡x\log x is one operator applied to xx — not l⋅o⋅g⋅xl \cdot o \cdot g \cdot x, and the subscript multiplies nothing.
log⁡x\log x
Log of x — base depends on who is writing
The unsubscripted log is a dialect. School texts, engineering, and this page read base 1010; advanced mathematics — analysis, number theory — reads base ee; computer science reads base 22. The software side of the split is mapped in Calculator Conventions below.
CasesWithin one document the convention is fixed and usually silent — the danger is crossing documents: a formula lifted from an analysis text into an engineering context silently changes by the constant factor ln⁡10\ln 10.
Also writtenlg⁡x\lg x — base 10 in Russian and European texts (the form this site's calculus pages may meet in imported formulas); lb⁡x\operatorname{lb} x — the ISO mark for base 2; ld⁡x\operatorname{ld} x — the older German logarithmus dualis.
Do not confuseA portable symbol. The three readings differ by constant factors, so graphs, slopes, and constants shift with the dialect — the formula looks identical and is not.
ln⁡x\ln x
Ell-en of x; natural log of x
Latin, not English: logarithmus naturalis — which is why the letters run “ln” and not “nl”. The mark is credited to Irving Stringham, 1893. Base ee always, no dialect, no exceptions — the one logarithm symbol that means the same thing everywhere.
CasesPowers attach to the operator the way trigonometry writes them: ln⁡2x\ln^2 x means (ln⁡x)2(\ln x)^2, borrowing the squared-function convention — while the iterate is spelled out as ln⁡ln⁡x\ln \ln x.
Also writtenlog⁡ex\log_e x, on the rare occasions the base is spelled; log⁡x\log x itself, in the analysis dialect above.
Do not confuseln⁡2x\ln^2 x with ln⁡(x2)\ln(x^2). The first squares the output, the second doubles it — ln⁡(x2)=2ln⁡x\ln(x^2) = 2\ln x by the logarithm rules — and the two agree only at x=1x = 1 and x=e2x = e^2.

The Natural Logarithm

The natural logarithm is the logarithm with base ee, where e≈2.71828e \approx 2.71828. It is written ln⁡(x)\ln(x), read "natural log of xx," or occasionally log⁡e(x)\log_e(x).

The number ee is not chosen arbitrarily. It emerges from the study of continuous growth: if a quantity grows at a rate proportional to its current size, ee appears in the formula describing that growth. Compound interest computed with infinitely many compounding periods, radioactive decay, population dynamics — all involve ee naturally.

In calculus, ee has a unique property: the derivative of exe^x is exe^x itself. No other base produces this self-replicating behavior under differentiation. The natural logarithm inherits a corresponding property: the derivative of ln⁡(x)\ln(x) is 1/x1/x, the simplest possible form.

The values ln⁡(1)=0\ln(1) = 0 and ln⁡(e)=1\ln(e) = 1 follow from the general properties. Additional reference points: ln⁡(2)≈0.693\ln(2) \approx 0.693, ln⁡(10)≈2.303\ln(10) \approx 2.303. These approximations appear frequently in applications.

The Number e

The constant ee is an irrational number approximately equal to 2.718282.71828. Its decimal expansion continues without repeating: e=2.718281828459045...e = 2.718281828459045...

One definition comes from compound interest. If \1earns earns 100\%interestperyear,compounded interest per year, compounded ntimes,theyear−endbalanceis times, the year-end balance is (1 + 1/n)^n.As. As nincreases—daily,hourly,everysecond,continuously—thisexpressionapproaches increases — daily, hourly, every second, continuously — this expression approaches e$:

e=lim⁡n→∞(1+1n)ne = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n


Another definition uses an infinite series:

e=1+1+12!+13!+14!+⋯=∑n=0∞1n!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\log_e to the status of "natural."

Three independent definitions of ee 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⁡\log for the common logarithm (base 1010) and ln⁡\ln for the natural logarithm (base ee). 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 math.log(x)\texttt{math.log(x)} function computes the natural logarithm by default, not the common logarithm. The common logarithm requires math.log10(x)\texttt{math.log10(x)}. Other languages follow similar patterns — always check documentation.

Spreadsheet software like Excel uses LOG(x)\texttt{LOG(x)} for common logarithm and LN(x)\texttt{LN(x)} for natural logarithm, matching calculator conventions. The function LOG(x, base)\texttt{LOG(x, base)} allows arbitrary bases.

When in doubt, test with known values. If the software returns 11 for log⁡(10)\log(10), it uses base 1010. If it returns 11 for log⁡(e)\log(e), it uses base ee.

The log calculator has separate options for base 10, base e and base 2, plus a custom base, so you do not have to rely on these conventions.
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 1010. A change of 11 on these scales represents a factor of 1010 — intuitive for decimal-system thinkers.

Use natural logarithms when working with continuous growth or decay, calculus, or theoretical derivations. Exponential models of the form AektAe^{kt} pair naturally with ln⁡\ln. Solving e2x=5e^{2x} = 5 is cleanest with natural logarithms: 2x=ln⁡(5)2x = \ln(5), so x=ln⁡(5)/2x = \ln(5)/2.

For solving general exponential equations, either logarithm works. The equation 3x=73^x = 7 can be solved as x=log⁡(7)/log⁡(3)x = \log(7)/\log(3) or x=ln⁡(7)/ln⁡(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)=log⁡b(x)log⁡b(a)\log_a(x) = \frac{\log_b(x)}{\log_b(a)}


To convert a logarithm of any base to common logarithms:

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


To convert to natural logarithms:

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


For example, log⁡2(10)=ln⁡(10)ln⁡(2)=2.3030.693≈3.322\log_2(10) = \frac{\ln(10)}{\ln(2)} = \frac{2.303}{0.693} \approx 3.322. Verification: 23.322≈102^{3.322} \approx 10.

Converting between common and natural logarithms directly uses ln⁡(10)≈2.303\ln(10) \approx 2.303:

ln⁡(x)=log⁡(x)⋅ln⁡(10)≈2.303⋅log⁡(x)\ln(x) = \log(x) \cdot \ln(10) \approx 2.303 \cdot \log(x)


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


These conversions allow computation with any base using only the log⁡\log and ln⁡\ln buttons available on standard calculators.

Comparing Graphs

The graphs of y=log⁡(x)y = \log(x) and y=ln⁡(x)y = \ln(x) share the same basic shape but differ in steepness.

Both pass through (1,0)(1, 0) — all logarithms satisfy log⁡a(1)=0\log_a(1) = 0. The common logarithm passes through (10,1)(10, 1); the natural logarithm passes through (e,1)≈(2.718,1)(e, 1) \approx (2.718, 1). Both have vertical asymptotes at x=0x = 0 and extend to +∞+\infty as xx increases.

Since e<10e < 10, the natural logarithm reaches 11 at a smaller input value. This makes ln⁡(x)\ln(x) grow faster initially and appear steeper for small xx. For any positive x>1x > 1, ln⁡(x)>log⁡(x)\ln(x) > \log(x) because the natural logarithm uses a smaller base.

The ratio between them is constant: ln⁡(x)=ln⁡(10)⋅log⁡(x)≈2.303⋅log⁡(x)\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)\ln(10) produces the natural logarithm.
ln x and log x: both through (1, 0); ln x is log x stretched by ln 10xy123456789101112−2−1123ln xlog x(e, 1)(10, 1)(10, ln 10 ≈ 2.303)at every x: ln x ≈ 2.303 · log xSame shape: ln x is log x stretched by ln 10 ≈ 2.303
One curve, two vertical scales

ln x and log x drawn together. Both cross the axis at (1, 0). The natural logarithm reaches 1 already at x = e ≈ 2.718, the common logarithm only at x = 10, so ln x is the steeper of the two. At x = 10 the dashed line shows the fixed ratio between them: log 10 = 1 while ln 10 ≈ 2.303, and the same factor 2.303 links their heights at every x.

Every logarithm is a vertical stretch of every other, by the change-of-base constant.

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)

Common and Natural Logarithms FAQ

What base does a bare log⁡\log mean?

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It depends on the field, and the three readings disagree. School texts and engineering read base 1010; analysis and number theory read base ee; computer science reads base 22. Within one document the convention is fixed and usually left unstated. The hazard is crossing documents, where a formula lifted from an analysis text into an engineering context silently shifts by a factor of ln⁡10\ln 10.Read more →

Does log⁡bx+1\log_b x + 1 mean log⁡b(x+1)\log_b(x + 1)?

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No. Like sin⁡x\sin x, the logarithm binds only to what immediately follows it, so log⁡bx+1\log_b x + 1 reads as (log⁡bx)+1(\log_b x) + 1. At x=9x = 9 in base 1010 that is about 1.9541.954, while log⁡10(9+1)\log_{10}(9 + 1) is exactly 11. Once the argument is a sum the parentheses are obligatory.Read more →

What does ln⁡2x\ln^2 x mean?

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It means (ln⁡x)2(\ln x)^2, the output squared. The superscript attaches to the operator, borrowing the squared-function convention from trigonometry, where sin⁡2x\sin^2 x behaves the same way. Repeated application gets spelled out instead, as ln⁡ln⁡x\ln \ln x. Note that ln⁡2x\ln^2 x and ln⁡(x2)\ln(x^2) are different quantities: the second doubles rather than squares.Read more →

Why is the natural logarithm written ln and not nl?

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Because the letters are Latin rather than English: logarithmus naturalis, so the ll leads. The mark is credited to Irving Stringham in 1893. Unlike a bare log⁡\log, it carries no dialect at all, meaning base ee in every field, which makes it the one logarithm symbol you can move between documents without checking first.Read more →