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

Step-by-step solution

1. 1

   Spot where the unknown constant sits. Here $b$ is an exponent on the variable $x$, not on a constant. That difference decides the transformation: a power law needs logs on both sides and both variables, whereas an exponential law $y=ab^{x}$ only needs a log on $y$.

2. 2

   Take logarithms of both sides.

   $\log y=\log\!\left(a\,x^{b}\right)$

3. 3

   Split the product. By $\log(MN)=\log M+\log N$:

   $\log y=\log a+\log\!\left(x^{b}\right)$

4. 4

   Bring the power down. By $\log(x^{b})=b\log x$:

   $\log y=b\log x+\log a$

5. 5

   Match against $Y=mX+c$.

   $\underbrace{\log y}_{Y}=\underbrace{b}_{m}\underbrace{\log x}_{X}+\underbrace{\log a}_{c}$

   Plotting $\log y$ against $\log x$ - a log-log plot - gives a straight line whose gradient is the exponent $b$ and whose intercept is $\log a$.

6. 6

   Verify with numbers. Take $a=3,b=2,x=4$, so $y=3\cdot 16=48$. Then $\log_{10}48=1.6812$, while $b\log_{10}4+\log_{10}3=2(0.6021)+0.4771=1.6812$ ✓. Recovering the constants: $b=m$ and $a=10^{c}$.