Function library
Natural Logarithm Function
f(x) = log(x)
The inverse of eˣ — it unwraps exponential growth, turning multiplication into addition.
The natural logarithm, written ln(x) or log(x), answers the question “e to what power gives x?” It is the exact inverse of the exponential function: ln(eˣ) = x and e^(ln x) = x, so its graph is the mirror image of y = eˣ reflected across the line y = x. Where the exponential rockets upward, the logarithm climbs with agonizing slowness — ln(10) ≈ 2.303, ln(100) ≈ 4.605, ln(1,000,000) ≈ 13.816 — each tenfold increase in x adds only about 2.303 to the output.
That slow growth is precisely the point: logarithms compress enormous ranges into manageable ones, which is why the Richter scale, decibels, and pH are all logarithmic. The graph passes through (1, 0), rises for x > 1, dives to −∞ as x approaches 0 from the right (the vertical asymptote at x = 0), and is undefined for x ≤ 0 — you cannot raise e to any real power and get zero or a negative number. Type log(x) into the calculator alongside e^x to see the mirror symmetry.
Computed properties
Calculated by the graphing calculator's own math engine at build time for log(x).
- Roots in [−10, 10]
- 1.0000
- y-intercept f(0)
- undefined
- Local extrema in [−10, 10]
- none found
- Sample values
- f(-2) = undefined · f(-1) = undefined · f(0) = undefined · f(1) = 0 · f(2) = 0.6931
- Derivative f′(1)
- 1.0000
- Integral ∫₀¹ f(x) dx
- undefined
Key facts
- Domain: x > 0; range: all real numbers. Undefined at x ≤ 0.
- Inverse of eˣ: ln(eˣ) = x and e^(ln x) = x; graphs mirror across y = x.
- x-intercept at (1, 0); vertical asymptote x = 0 with ln(x) → −∞ as x → 0⁺.
- Log laws: ln(ab) = ln a + ln b; ln(a/b) = ln a − ln b; ln(a^b) = b·ln a.
- Derivative d/dx ln(x) = 1/x; ln is the antiderivative of 1/x.
- Increasing and concave down on its whole domain; no maxima, minima, or inflection points.
What the natural logarithm is
Logarithms were invented to turn multiplication into addition: ln(ab) = ln(a) + ln(b). Before electronic calculators, scientists multiplied large numbers by looking up their logarithms, adding, and converting back — the slide rule is a physical embodiment of this idea. The “natural” in the name refers to base e, the base that makes calculus clean: d/dx ln(x) = 1/x, the simplest possible derivative for an inverse-exponential.
The three log laws follow from the exponent laws of its inverse: ln(ab) = ln a + ln b, ln(a/b) = ln a − ln b, and ln(a^b) = b·ln a. Together they let you dismantle complicated multiplicative expressions into sums — the reason logarithms appear in entropy formulas, likelihood calculations, and anywhere products become unwieldy.
Domain, asymptote, and shape
The domain is x > 0 only, a direct consequence of eˣ > 0: there is no real power of e that yields zero or a negative number, so the logarithm cannot accept them. As x → 0⁺, ln(x) → −∞, giving the vertical asymptote x = 0 (the y-axis) — the mirror of the exponential’s horizontal asymptote. The x-intercept is at (1, 0) since e^0 = 1, the mirror of eˣ’s y-intercept at (0, 1).
The curve is increasing everywhere (derivative 1/x > 0 for x > 0) but concave down everywhere (second derivative −1/x² < 0): it rises quickly just right of zero, then flattens relentlessly. It has no maximum and no inflection point, and it is the antiderivative of 1/x — the integral that no power rule can handle, since ∫xⁿ dx fails at n = −1.
Where logarithms appear
Logarithmic scales measure phenomena spanning many orders of magnitude: each Richter point is about 32× the energy, each pH unit is 10× the acidity, and decibels compress sound intensities from a whisper to a jet engine into a 0–140 range. In information theory, entropy is measured in nats (natural log) or bits (log base 2), quantifying surprise and optimal code lengths.
In computer science, O(log n) algorithms — binary search being the classic — halve the problem each step, so doubling the input adds just one more step; that is logarithmic growth in action. In statistics, taking logs straightens exponential data into lines, and the log-normal distribution models quantities like incomes and particle sizes that multiply rather than add.
Frequently asked questions
Why is ln(x) undefined for negative x?
Because ln(x) asks “e to what power equals x?”, and e raised to any real power is always positive. No real exponent produces zero or a negative number, so the logarithm has no real value there. (Complex logarithms exist but are multi-valued and beyond this real-valued graph.)
What is the difference between ln(x) and log₁₀(x)?
Only the base: ln uses e ≈ 2.718, log₁₀ uses 10. They are proportional — ln(x) = ln(10)·log₁₀(x) ≈ 2.303·log₁₀(x) — so their graphs have identical shapes, just different vertical scales. Natural logs give the cleanest calculus (derivative 1/x); base-10 logs suit decimal-scale measurements.
Why does the graph flatten out so much?
Because undoing exponential growth is inherently slow: to increase ln(x) by 1 you must multiply x by e ≈ 2.718. The derivative 1/x shrinks as x grows, so each additional unit of height requires an ever-larger multiple of x. That flattening is exactly what makes logarithms ideal for compressing huge ranges.