Find the length of the polar subtangent and the polar

Find the length of the polar subtangent and the polar subnormal of the logarithmic spiral $\rho=a^{\varphi}$, $a>0,\ a\neq1$.

Step 1: The two formulas

\[ S_T = \left|\frac{\rho^2}{\rho'(\varphi)}\right|, \qquad S_N = \left|\rho'(\varphi)\right|. \tag{1} \]

Step 2: Differentiate

\[ \rho' = a^{\varphi}\ln a = \rho\ln a. \tag{2} \]

Step 3: Subtangent

\[ S_T = \left|\frac{\rho^2}{\rho\ln a}\right| = \frac{\rho}{|\ln a|} = \frac{a^{\varphi}}{|\ln a|}. \tag{3} \]

Step 4: Subnormal

\[ S_N = |\rho\ln a| = \rho\,|\ln a| = a^{\varphi}\,|\ln a|. \tag{4} \]

Step 5: A companion identity

Multiplying (3) and (4): \[S_T\cdot S_N = \frac{\rho}{|\ln a|}\cdot\rho|\ln a| = \rho^2, \tag{5}\] so the radius vector is the geometric mean of the polar subtangent and the polar subnormal, at every point of the spiral.

Step 6: Verification

\[a=1.5,\ \varphi=2.0\text{: } \rho\approx2.25000,\ S_T\approx5.54898,\ S_N\approx0.91259,\ S_T S_N\approx5.06250=\rho^2\ \checkmark\]