Logarithmic Spiral — Definition, Formula & Examples

A logarithmic spiral is a curve that winds around a central point while moving farther away from it, with the distance from the center growing exponentially as the angle increases. It appears naturally in seashells, hurricanes, and galaxies.

A logarithmic spiral is a curve defined in polar coordinates by $r = ae^{b\theta}$ , where $a > 0$ is the initial radius (the value of $r$ when $\theta = 0$ ) and $b \neq 0$ is a constant controlling how tightly the spiral winds. A distinguishing geometric property is that the angle between the tangent line and the radial line at every point is constant.

Key Formula

r = ae^{b\theta}

Where:

$r$ = Distance from the origin (pole) to a point on the spiral
$a$ = Positive constant equal to the radius when θ = 0
$b$ = Nonzero constant controlling the rate and direction of expansion
$\theta$ = Angle in radians measured from the positive x-axis

How It Works

To graph a logarithmic spiral, choose values of

a

and

b

, then compute

r

for several values of

\theta

. When

b > 0

, the spiral expands as

\theta

increases (counterclockwise). When

b < 0

, it contracts instead. A larger

|b|

makes the spiral spread out faster, while a smaller

|b|

produces a tighter coil. The constant-angle property—called the equiangular property—means every radial ray from the origin crosses the spiral at the same angle.

Worked Example

Problem: Plot several points on the logarithmic spiral r = 2e^(0.1θ) for θ = 0, π, 2π, and 3π.

Step 1: Substitute θ = 0 into the equation.

r = 2e^{0.1(0)} = 2e^{0} = 2

Step 2: Substitute θ = π ≈ 3.14.

r = 2e^{0.1\pi} \approx 2e^{0.314} \approx 2(1.369) \approx 2.74

Step 3: Substitute θ = 2π and θ = 3π.

r = 2e^{0.2\pi} \approx 3.75, \quad r = 2e^{0.3\pi} \approx 5.14

Answer: The points (r, θ) are approximately (2, 0), (2.74, π), (3.75, 2π), and (5.14, 3π). Each full revolution increases the radius by a factor of e^(0.2π) ≈ 1.87, confirming exponential growth.

Why It Matters

Logarithmic spirals model growth patterns in biology (nautilus shells, sunflower seeds) and physics (spiral galaxies, cyclone arms). In precalculus and calculus courses, they serve as a key example of polar curves and exponential behavior combined, and they appear again in complex analysis and differential equations.

Common Mistakes

Mistake: Confusing the logarithmic spiral with the Archimedean spiral r = a + bθ, which grows linearly rather than exponentially.

Correction: Check the equation: if r depends on e^(bθ), it is logarithmic; if r depends on θ directly (no exponential), it is Archimedean. The logarithmic spiral's spacing between successive turns increases, while the Archimedean spiral's spacing stays constant.