Your equation is:

U_{i,t}=\left[q_{1}^{i} \frac{C_{i,t}^{1-\eta_{i}}}{1-\eta_{i}}-\frac{N_{i,t}^{1+\theta_{i}}}{1+\theta_{i}}+(1-q_{1}^{i})(G_{i,t})\right]

Doing Uhlig’s substitution X_t = Xe^{\tilde{X}_t}, where \tilde{X}_t =\ln{X_t/X} (note that dropping t subscript means steady state), yield:

U_{i,t}=\left[q_{1}^{i} \frac{(C_{i}e^{\tilde{C}_{i,t}})^{1-\eta_{i}}}{1-\eta_{i}}-\frac{(N_{i}e^{\tilde{N}_{i,t}})^{1+\theta_{i}}}{1+\theta_{i}}+(1-q_{1}^{i})(G_{i}e^{\tilde{G}_{i,t}})\right]

U_{i,t}=\left[q_{1}^{i} \frac{C_{i}^{1-\eta_{i}}e^{(1-\eta_i)\tilde{C}_{i,t}}}{1-\eta_{i}}-\frac{N_{i}^{1+\theta_{i}}e^{(1+\theta_{i})\tilde{N}_{i,t}}}{1+\theta_{i}}+(1-q_{1}^{i})(G_{i}e^{\tilde{G}_{i,t}})\right]

From this point we perform the approximation by acknowledging that, up to first order: e^{a\tilde{X}_t}\approx 1+a\tilde{X}_t. Replacing this approximation gives:

U_{i,t}=\left[q_{1}^{i} \frac{C_{i}^{1-\eta_{i}}(1+(1-\eta_i)\tilde{C}_{i,t})}{1-\eta_{i}}-\frac{N_{i}^{1+\theta_{i}}(1+(1+\theta_{i})\tilde{N}_{i,t})}{1+\theta_{i}}+(1-q_{1}^{i})(G_{i}(1+\tilde{G}_{i,t}))\right]

(more precisely

U_{i,t}\approx\left[q_{1}^{i} \frac{C_{i}^{1-\eta_{i}}(1+(1-\eta_i)\tilde{C}_{i,t})}{1-\eta_{i}}-\frac{N_{i}^{1+\theta_{i}}(1+(1+\theta_{i})\tilde{N}_{i,t})}{1+\theta_{i}}+(1-q_{1}^{i})(G_{i}(1+\tilde{G}_{i,t}))\right]

)

From here the expression is already log-linearized, clearly you can do some algebra to simplify things out and express it more cleanly.