The relation y p^x = q is non-linear. Change it into linear form Y = mX + c and state Y, X, m and c.

Step-by-step solution

1. 1

   Decide which transformation the equation calls for. The unknown $x$ sits in an exponent, so no amount of rearranging in $x$ and $y$ will straighten the graph. The tool that pulls a variable down out of an exponent is the logarithm - that is the whole strategy.

2. 2

   Isolate $y$ first. Dividing by $p^{x}$:

   $y=\frac{q}{p^{x}}=q\,p^{-x}$

3. 3

   Take logarithms of both sides. Any base works as long as it is used consistently:

   $\log y=\log\!\left(q\,p^{-x}\right)$

4. 4

   Split the product and drop the exponent. Using $\log(AB)=\log A+\log B$ and $\log(p^{-x})=-x\log p$:

   $\log y=\log q-x\log p$

5. 5

   Match it against $Y=mX+c$. Rewriting with the $x$-term first:

   $\underbrace{\log y}_{Y}=\underbrace{(-\log p)}_{m}\underbrace{x}_{X}+\underbrace{\log q}_{c}$

   So you plot $\log y$ against $x$ itself - not against $\log x$, which is the usual slip.

6. 6

   Check the reading of the graph. The gradient is $-\log p$, so $p=10^{-m}$ for base-10 logs; the intercept is $\log q$, so $q=10^{c}$. A numerical test with $p=2,q=5,x=3$: $y=5/8=0.625$, and $\log_{10}0.625=-0.2041$, while $-3\log_{10}2+\log_{10}5=-0.9031+0.6990=-0.2041$ ✓.