Figure 13.17 showed a graph of log T versus log r for the planetary data given in Table 13.2. Such a graph is called a log-log graph. The scales in Figure 13.17 are logarithmic, not linear, meaning
that each division along the axis corresponds to a factor of 10 increase in the value. Strictly speaking, the “correct” labels on the y-axis should be 7, 8, 9, and 10 because these are the logarithms of 10⁷...... 10¹⁰.
a. Consider two quantities u and v that are related by the expression v_p = Cu^q, where C is a constant. The exponents p and q are not necessarily integers. Define x = log u and y = log v. Find
an expression for y in terms of x.
b. What shape will a graph of y versus x have? Explain.
c. What slope will a graph of y versus x have? Explain.
d. Use the experimentally determined “best-fit” line in Figure 13.17 to find the mass of the sun.

the expression v_p = Cu^q, where C is a constant. The exponents p and q are not necessarily integers. Define x = log u and y = log v.

From the question we have expression
$v^p=C{u}^q$

Take log on both side

$$\begin{gathered} \log \left(v^p\right)=p\log v \\ \log \left(C{u}^q\right)=\log C+q\log u \\ p\log v=\log C+q\log u \\ \log v=\frac{\log C}{p}+\frac{q}{p}\log u \end{gathered}$$

From the above equation we can say that

$y=\left(\frac{q}{p}\right)x+\frac{\log C}{p}$

This is the equation

the expression v_p = Cu^q, where C is a constant. The exponents p and q are not necessarily integers. Define x = log u and y = log v.

and the graph in figure.

In part(a) we have established a relationship. Which is a linear variation in the form of y=mx+c. hence, there is a linear relationship between log u and log v.

So, the graph between log v versus log u is a straight line.

the expression v_p = Cu^q, where C is a constant. The exponents p and q are not necessarily integers. Define x = log u and y = log v.

and the graph in figure.

In the part(a) of the problem we have established relationship between $y and x as

$y=\left(\frac{q}{p}\right)x+\frac{\log C}{p}$

and the shape of the graph is a straight line in the form of y= mx +c

Where, m = slope of the graph and c= constant.
On comparing similar parts of both equation we get

$$\begin{gathered} m=\frac{q}{p} \\ c=\frac{\log C}{p} \end{gathered}$$

An the slope is q/p

the expression v_p = Cu^q, where C is a constant. The exponents p and q are not necessarily integers. Define x = log u and y = log v.

and the graph in figure.

We will compare the Kepler's third law of planetary motion with the given equation

and use the slope as calculated.

$T^2=\left(\frac{4R^3}{GM}\right)\mathrm{p}$

Calculate slope

$\frac{q}{p}=\frac{3}{2}=1.5$

now

$\frac{1}{2}\log C=\frac{1}{2}\log \left(4{\pi }^2/G{M}_{sun}\right)=-9.264$

Solve the equation with respect to base 10

$$\begin{gathered} M_{\text{sun}}=\left(4{\pi }^2/G\right)\times {10}^{18.528} \\ {M}_{\text{sun}}=1.996\times {10}^{30}\mathrm{kg} \end{gathered}$$

Approx mass is 2 x 10³⁰ kg