A straight line passes through the point $P(\log_2 16 , \log_3 27)$ such that the portion of the line intercepted between the co-ordinate axes is divided by $P$ in the ratio $1:2$ internally (starting from the $x$-axis). Then the equation of the line is:

The coordinates of the point $P$ are given by $(\log_2 16, \log_3 27)$.

Since $\log_2 16 = \log_2 (2^4) = 4$ and $\log_3 27 = \log_3 (3^3) = 3$, the point is $P(4, 3)$.

Let the line intersect the $x$-axis at $A(a, 0)$ and the $y$-axis at $B(0, b)$.

It is given that $P(4, 3)$ divides the line segment $AB$ internally in the ratio $1:2$. Using the section formula, the coordinates of $P$ are:

$x = \dfrac{1(0) + 2(a)}{1 + 2} = \dfrac{2a}{3}$

$y = \dfrac{1(b) + 2(0)}{1 + 2} = \dfrac{b}{3}$

Equating these to the coordinates of $P(4, 3)$:

$\dfrac{2a}{3} = 4 \Rightarrow 2a = 12 \Rightarrow a = 6$

$\dfrac{b}{3} = 3 \Rightarrow b = 9$

The intercepts of the line are $a = 6$ and $b = 9$. The equation of the line in intercept form is:

$\dfrac{x}{a} + \dfrac{y}{b} = 1$

$\dfrac{x}{6} + \dfrac{y}{9} = 1$

Multiplying the entire equation by $18$, we get:

$3x + 2y = 18$

$3x + 2y - 18 = 0$

Answer: $3x + 2y - 18 = 0$