= Solution
Use the <scattering potential> convention $V=k_0^2(n^2-1)$. With time dependence $e^{-i\omega t}$ the outgoing <Green function> is $G_0(r,r')=e^{ik_0|r-r'|}/(4\pi|r-r'|)$ and satisfies $(\Delta+k_0^2)G_0=-\delta$. The total-field <Lippmann-Schwinger equation> is
$$
\psi(r)=\psi_0(r)+\int_DG_0(r,r')V(r')\psi(r')\,d^3r'.
$$
Replacing the field in the integral by the incident field gives the first <Born approximation for scalar wave scattering>:
$$
\boxed{\psi_B=\psi_0+\psi_1,\qquad \psi_1(r)=\int_DG_0(r,r')V(r')\psi_0(r')\,d^3r'.}
$$
For the first <Rytov approximation>, write $\psi=\psi_0e^\chi$ on a region where the incident field is nonzero and a continuous <complex logarithm> can be chosen. The <logarithmic wave perturbation> equation is
$$
\Delta\chi+2\nabla\log\psi_0\cdot\nabla\chi+\nabla\chi\cdot\nabla\chi=-V.
$$
The last dot product is bilinear, not a squared modulus. Dropping it leaves a linear equation for $\chi_1$. Multiplication by $\psi_0$ shows that $\psi_0\chi_1$ satisfies the same outgoing inhomogeneous <Helmholtz equation> as $\psi_1$. Hence
$$
\boxed{\chi_1=\frac{\psi_1}{\psi_0},\qquad \psi_R=\psi_0\exp\left(\frac{\psi_1}{\psi_0}\right).}
$$
Introduce a small multiplier $\lambda$ on $V$. Then $\psi_1=O(\lambda)$ and
$$
\psi_R=\psi_0+\psi_1+\frac{\psi_1^2}{2\psi_0}+O(\lambda^3)=\psi_B+O(\lambda^2).
$$
\b[The two total-field approximations agree to first order.] Exponentiating the first <logarithmic wave perturbation> does not supply the generally missing second-order <Born series> contribution.
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