The general procedure is to plot the data on log--log paper (old way), to plot $\log(\text{RHS})$ versus $\log(\text{LHS})$ on normal paper (alternate old way, supported on spreadsheets) or just hand the data to a robust minimization package and ask it for a power-law fit (new way).
The first two methods are equivalent and the work because
$$
\begin{array}
{} y &= bx^a \\
\log(y) &= \log(bx^a) \\
\log(y) &= \log(x^a) + \log(b) \\
\log(y) &= a \log(x) + \log(b)
\end{array}
$$
shows that the curve is transformed into a line---which is easy to fit with a ruler or a spreadsheet---where the power is the slope and the intercept is the logarithm of the coefficient.
By the way, the word "exponential" is generally reserved for relations like $y = \exp(x)$, which is managed in the old way by plotting on semi-log paper. You want "power-law".