Linearising it by log-transforming the data is not appropriate, because that distorts the errors, making them multiplicative rather than additive.
It is straightforward to do a nonlinear parameter estimation using fminsearch, which is a core-MATLAB function, requiring no toolboxes.
EDIT — (8 Sep 2021 at 15:30)
To illustrate —
y = x.^2 + randn(size(x));
objfcn = @(b,x) b(1).*exp(b(2).*x);
[B1,Fval] = fminsearch(@(b) norm(y - objfcn(b,x)), B0)
B2 = polyfit(x, log(y), 1)