We are tasked with identifying the equivalent form of the function $f(x)$ that displays $m$ as a constant or a coefficient, given that for each increase in $x$ by $\frac{1}{4}$, the value of $f(x)$ increases by a factor of $m$.

The general form of an exponential function is:

$$f(x) = a \cdot m^{nx}$$

From the question, we know that:

• When $x$ increases by $\frac{1}{4}$, $f(x)$ is multiplied by $m$.
• This implies $f(x) = 13 \cdot m^{4x}$, as $4x$ ensures the multiplication factor $m$ occurs every $\frac{1}{4}$ increase in $x$.

Given that $m = 10$ (as shown in the calculations), the function becomes:

$$f(x) = 13 \cdot (10)^{4x}$$

The equivalent form of the function can be written as:

$$f(x) = 13 \cdot (10^{3/2})^{6x}$$

This matches option C.

Conclusion:

The equivalent form displaying $m$ as a constant or coefficient is option C.

Answer: C) $f(x) = 13 \cdot (10^{3/2})^{6x}$