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Ln (Natural Logarithm) — Definition, Formula & Examples

Ln (natural logarithm) is the logarithm with base e≈2.718e \approx 2.718, written as ln⁡(x)\ln(x). It answers the question: "To what power must ee be raised to produce xx?"

For any positive real number xx, the natural logarithm ln⁡(x)\ln(x) is defined as the unique real number yy such that ey=xe^y = x. Equivalently, ln⁡(x)=log⁡e(x)\ln(x) = \log_e(x). The function ln⁡:(0,∞)→R\ln: (0, \infty) \to \mathbb{R} is the inverse of the exponential function f(t)=etf(t) = e^t.

Key Formula

y=ln⁡(x)  ⟺  ey=xy = \ln(x) \iff e^y = x
Where:
  • xx = A positive real number (the input to ln)
  • yy = The exponent to which e must be raised to equal x
  • ee = Euler's number, approximately 2.71828

How It Works

When you see ln⁡(x)\ln(x), you are finding the exponent that turns ee into xx. For instance, ln⁡(e3)=3\ln(e^3) = 3 because ee raised to the 3rd power gives e3e^3. The natural logarithm converts multiplication into addition — ln⁡(ab)=ln⁡(a)+ln⁡(b)\ln(ab) = \ln(a) + \ln(b) — which makes it powerful for simplifying products and solving exponential equations. To solve an equation like e2t=50e^{2t} = 50, take ln⁡\ln of both sides to bring the variable out of the exponent: 2t=ln⁡(50)2t = \ln(50).

Worked Example

Problem: Solve for tt: e2t=20e^{2t} = 20.
Step 1: Take the natural logarithm of both sides to remove the exponential.
ln⁡(e2t)=ln⁡(20)\ln(e^{2t}) = \ln(20)
Step 2: Use the property ln⁡(ek)=k\ln(e^k) = k to simplify the left side.
2t=ln⁡(20)2t = \ln(20)
Step 3: Divide both sides by 2 and evaluate with a calculator.
t=ln⁡(20)2≈2.9962≈1.498t = \frac{\ln(20)}{2} \approx \frac{2.996}{2} \approx 1.498
Answer: t≈1.498t \approx 1.498

Another Example

Problem: Simplify ln⁡(e5⋅e−2)\ln(e^5 \cdot e^{-2}).
Step 1: Combine the exponentials using exponent rules.
ln⁡(e5⋅e−2)=ln⁡(e5+(−2))=ln⁡(e3)\ln(e^5 \cdot e^{-2}) = \ln(e^{5+(-2)}) = \ln(e^3)
Step 2: Apply the inverse property: ln⁡(ek)=k\ln(e^k) = k.
ln⁡(e3)=3\ln(e^3) = 3
Answer: ln⁡(e5⋅e−2)=3\ln(e^5 \cdot e^{-2}) = 3

Visualization

Why It Matters

The natural logarithm is central to AP Calculus, where ln⁡(x)\ln(x) is the antiderivative of 1x\frac{1}{x} and appears in integration techniques throughout the course. In science and finance, ln is used to model continuous growth and decay — for example, calculating the half-life of a radioactive substance or the continuously compounded return on an investment. Any field that uses the exponential function exe^x relies on ln⁡\ln as its inverse to solve for unknown rates or times.

Common Mistakes

Mistake: Thinking ln⁡(a+b)=ln⁡(a)+ln⁡(b)\ln(a + b) = \ln(a) + \ln(b).
Correction: The logarithm of a sum does NOT split. The correct product rule is ln⁡(a⋅b)=ln⁡(a)+ln⁡(b)\ln(a \cdot b) = \ln(a) + \ln(b). There is no simple rule for ln⁡(a+b)\ln(a + b).
Mistake: Writing ln⁡(0)=0\ln(0) = 0 or trying to evaluate ln⁡\ln of a negative number.
Correction: The natural logarithm is only defined for positive inputs. ln⁡(x)→−∞\ln(x) \to -\infty as x→0+x \to 0^+, and ln⁡(1)=0\ln(1) = 0 (not ln⁡(0)\ln(0)). There is no real-number output for ln⁡(0)\ln(0) or ln⁡(−3)\ln(-3).

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