Define $\hat{y}=\log\frac{y_t}{y}$ which means percentage deviation from steady state.
Then $\hat{y}=s_c \hat{c}+s_i \hat{i} + s_g \hat{g}$ where $s_c +s_i +s_g =1$.
I can't derive that result.
Is that $\log s_c \approx s_c$?
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At every time period, we have: $$ y_t = c_t + i_t + g_t $$ Long run steady state gives: $$ y = c + i + g $$ so, taking differences we have: $$ \begin{align*} &y_t - y = c_t - c + i_t - i + g_t - g,\\ \iff &\frac{y_t - y}{y} = \frac{c}{y}\frac{c_t - c}{c} + \frac{i}{y}\frac{i_t - i}{i}+ \frac{g}{y} \frac{g_t - g}{g} \end{align*} $$ Set $s_c = \dfrac{c}{y}, s_i = \dfrac{i}{y}$ and $s_g = \dfrac{g}{y}$ to be the shares of consumption, investment and government expenditures.
This gives: $$ \frac{y_t - y}{y} = s_c \frac{c_t - c}{c} + s_i \frac{i_t - i}{i} + s_g \frac{g_t - g}{g} $$ Now, we approximate the growth rates. For a variable $x_t$, consider a Taylor expansion of $\ln(x_t)$ around the steady state $x$ $$ \ln(x_t) \approx \ln(x) + \frac{(x_t - x)}{x} $$ This gives $\dfrac{x_t - x}{x} \approx \ln(x_t) - \ln(x) = \ln \dfrac{x_t}{x}$. Substituting gives: $$ \begin{align*} &\ln\frac{y_t}{y} \approx s_c \ln \frac{c_t}{c} + s_i \ln \frac{i_t}{i} + s_g \ln \frac{g_t}{g}.\\ \iff &\hat y = s_c \hat c + s_i \hat i + s_g \hat g. \end{align*} $$
