Transform the above equation into a straight line graph to determine Ro and A.

The equation provided is:

$$R = R_0 t^{0.7\lambda}$$

To transform this power law into a linear form, we take the natural logarithm ($\ln$) of both sides of the equation:

$$\ln(R) = \ln(R_0 t^{0.7\lambda})$$

Using the logarithm property $\ln(ab) = \ln(a) + \ln(b)$, we can separate the terms on the right side:

$$\ln(R) = \ln(R_0) + \ln(t^{0.7\lambda})$$

Next, using the logarithm property $\ln(a^b) = b \ln(a)$, we can bring the exponent down:

$$\ln(R) = \ln(R_0) + (0.7\lambda) \ln(t)$$

We can rewrite the equation as:

$$\ln(R) = (0.7\lambda) \ln(t) + \ln(R_0)$$

Comparing this to the standard straight-line equation $y = mx + c$:

• $y = \ln(R)$
• $m = 0.7\lambda$ (the slope)
• $x = \ln(t)$
• $c = \ln(R_0)$ (the y-intercept)

To determine $R_0$ and $\lambda$ from a straight-line graph:

1. Plot $\ln(R)$ on the y-axis against $\ln(t)$ on the x-axis.
2. The graph will yield a straight line.
3. The slope of this line, $m$, will be equal to $0.7\lambda$. Therefore, $\lambda = \frac{m}{0.7}$.
4. The y-intercept of this line, $c$, will be equal to $\ln(R_0)$. Therefore, $R_0 = e^c$.