The natural logarithm of a positive number x, written lnx, is the power you must raise e ≈ 2.718 to in order to get x. So lnx = y means ey = x. For example, ln 1 = 0, lne = 1, and ln 20 ≈ 3.0 because e3 ≈ 20.1.
lnx and ex are inverse functions, so their graphs are reflections across y = x. Every point (a, b) on one matches (b, a) on the other, such as (1, e) and (e, 1).
Formal definition and notation
The natural logarithm is the logarithm with base e, where e = 2.71828… is an irrational number. For x > 0, lnx = logex is the unique real numbery with ey = x. The function y = lnx is the inverse of y = ex: ln (ex) = x for every real x, and elnx = x for every x > 0. Its domain is x > 0, its range is all real numbers, and it increases everywhere. In calculus, the slope of y = lnx at any point is exactly 1x.
An account balance grows by the model A = 500e0.04t, where t is in years. How long does it take to reach 1,000 dollars? Round to the nearest tenth of a year.
Logarithm is the general idea for any base b: logbx. The natural logarithm is the special case b = e.
Quick check: 3 questions
Question 1 of 3
What is ln (e7)?
Show answer and explanation
Answer: C. ln and e to a power undo each other, so ln (e7) = 7. e7 is the input itself, 7e multiplies instead of taking the exponent, and 1 is lne.
Question 2 of 3
Which equation means the same as lnx = 3?
Show answer and explanation
Answer: B.lnx = 3 means e3 = x, so x ≈ 20.09. 103 would come from the base-10 log, 3e multiplies, and ln 3 applies ln again instead of undoing it.
Question 3 of 3
Write ln 5 + ln 4 as a single logarithm.
Show answer and explanation
Answer: D. The product rule: ln 5 + ln 4 = ln (5 × 4) = ln 20. ln 9 adds the inputs, 20 drops the ln, and ln 1.25 is ln 5 − ln 4.
Natural Logarithm: frequently asked questions
What is the difference between ln and log?
Both are logarithms, but with different bases. ln is base e ≈ 2.718, and log on a calculator is base 10. So log 100 = 2 because 102 = 100, while ln 100 ≈ 4.605 because e4.605 ≈ 100. Some higher math books write log for ln, so check the context.
Can the natural log be negative?
Yes. The output of ln can be any real number. It is negative whenever the input is between 0 and 1: ln 0.5 ≈ −0.693, because e−0.693 ≈ 0.5. The input itself, however, must be positive. ln of 0 or of a negative number is undefined, since e to any power is positive.
What is ln 0?
ln 0 is undefined. There is no exponent that makes e to that power equal 0. As x gets closer to 0 from the right, lnx drops without bound, toward negative infinity, which is why the graph of y = lnx runs down along the y-axis without touching it.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSF-LE.A.4
Cite this entry
Effortless Math Team. “Natural Logarithm.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/natural-logarithm/
Flashcard
TermNatural Logarithmlnx
DefinitionThe logarithm with base e ≈ 2.718: lnx = y means ey = x. ln 1 = 0 and lne = 1.
LogarithmA logarithm is the exponent you need to raise a base to in order to get a given number. logbx = y means by = x, so log2 32 = 5 because 25 = 32. A log with no base written uses base 10: log 1000 = 3. Logs undo exponents.Functions · Algebra 2, PrecalculusIrrational NumberAn irrational number cannot be written as a fraction of two integers. Its decimal form does not end and does not repeat in a pattern. Examples are √2 = 1.41421… and π = 3.14159…. The square root of any whole number that is not a perfect square is irrational.Number System · Grade 8, Algebra 1Real NumberA real number is any number that can be placed on the number line. The real numbers are all the rational numbers (fractions, integers, terminating and repeating decimals) together with all the irrational numbers (such as √2 and π). Square roots of negative numbers, like √−4, are not real.Number System · Grade 8, Algebra 1DomainThe domain of a function is the set of all input values (usually x-values) the function can accept. Unless a problem says otherwise, it is every real number except values that would divide by zero or take an even root of a negative number. On a graph, the domain is how far the graph reaches left and right.Functions · Grade 8, Algebra 1, Algebra 2SlopeSlope is a number that measures how steep a line is: the change in y divided by the change in x between any two points on the line, often called rise over run. A positive slope rises from left to right, a negative slope falls, and a horizontal line has slope 0.Algebra · Grade 8, Algebra 1ExponentAn exponent tells how many times to use a base as a factor. It is the small raised number to the upper right of the base: in 53, the base is 5 and the exponent is 3, so 53 = 5 × 5 × 5 = 125. Any nonzero number with an exponent of 0 equals 1.Algebra · Grade 6, Grade 7, Grade 8PowerA power is a number written as a base with an exponent, such as 53, or the value that results, 53 = 5 × 5 × 5 = 125. We say 125 is the third power of 5, or “5 to the third power”. The second power is called the square and the third power the cube.Number System · Grade 5, Grade 6, Grade 7NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2UndefinedUndefined describes an expression that has no value, most often because it divides by zero. 70 is undefined, since no number times 0 gives 7. A vertical line has an undefined slope, and 3x − 2 is undefined at x = 2. Undefined does not mean zero: 07 = 0, but 70 has no value.Problem Solving · Grade 6, Grade 8, Algebra 1SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2QuotientA quotient is the answer you get when you divide one number by another. In 20 ÷ 4 = 5, the quotient is 5. The number being divided is the dividend and the number you divide by is the divisor. When the division is not exact, the quotient comes with a remainder, or as a decimal or fraction.Operations · Grade 3, Grade 4, Grade 5Negative NumberA negative number is a number less than zero, written with a minus sign, such as −3, −0.5 or −23. On a number line, negative numbers sit to the left of 0. They describe amounts below a starting point: temperatures below zero, depths below sea level, or money owed. Zero itself is neither negative nor positive.Number System · Grade 6, Grade 7InfinityInfinity is the idea of something without any end or bound, written with the symbol ∞. It is not a real number, so you cannot add, subtract or divide with it like an ordinary number. Math uses it for endless sets, quantities that grow without limit, and intervals such as (3, ∞).Symbols and Notation · Grade 6, Algebra 1, Algebra 2BaseA base is the number or expression that is repeatedly multiplied when using an exponent. In 45 the base is 4 and the exponent is 5, so 45 = 4 × 4 × 4 × 4 × 4 = 1024. The base is the big number written on the line; the exponent is the small raised number.Algebra · Grade 6, Grade 7, Grade 8FunctionA function is a rule that gives exactly one output for each input. If you put in one x-value, a function gives back one y-value. In f(x) = 2x + 1, the input 3 gives the output 7. Two inputs may share an output, but one input can never have two different outputs.Functions · Grade 8, Algebra 1, Algebra 2Euler's NumberEuler’s number, written e, is an irrational constant approximately equal to 2.71828. It is the number that ( 1 + 1n )n gets closer and closer to as n grows. It is the base of natural exponential growth, ex, and of the natural logarithm, lnx.Functions · Algebra 2, Precalculus, CalculusExponential GrowthExponential growth is an increase by the same percent, or the same factor, in each equal time period, so the amount grows faster and faster. It is modeled by y = a(1 + r)t, where a is the starting amount and r is the growth rate. Growing 10% a year multiplies the amount by 1.1 every year.Functions · Algebra 1, Algebra 2EquationAn equation is a statement that two expressions are equal. It always has an equal sign, such as x + 5 = 12 or 3a = 18. Solving an equation means finding the value of the variable that makes it true: here x = 7 and a = 6. An expression like 4x + 7 has no equal sign, so it is not an equation.Algebra · Grade 6, Grade 7, Grade 8Properties of LogarithmsThe properties of logarithms are rules for rewriting logs: the log of a product is a sum, logb(MN) = logbM + logbN; the log of a quotient is a difference; and the log of a power is a multiple, logb(Mp) = plogbM. They come from the rules of exponents.Functions · Algebra 2, PrecalculusExponential FunctionAn exponential function is a function in which the input is the exponent: f(x) = a · bx, with a ≠ 0, base b > 0 and b ≠ 1. Each time x goes up by 1, the output is multiplied by the same factor b. If b > 1 the function grows; if 0 < b < 1 it decays.Functions · Algebra 1, Algebra 2, Precalculus