Log-linear approximation

Hi all! I am not quite surethat log-linear approximation and a first order approximation mean the same thing? Then, equation like K_t=k1_t+k2_t need to do log-linear? Your kind assistance and response will be greatly appreciated.

There is a slight terminological confusion. Strictly speaking, a first order approximation just means that a first order Taylor approximation is conducted. It does not say anything whether this first order approximation is conducted for levels or log-levels (i.e. percentage deviations). A log-linear approximation in contrast means that a first order Taylor approximation for log-levels is conducted. If your equation in levels is

then it is already linear and a first order Taylor approximation in levels will return the same equations. A log-linear approximation in contrast would be:
K_barK_t_hat=k1_bark1_t_hat+k2_bar*k2_t_hat;
where bar denotes steady state of the original levels and hats denote percentage deviations from steady state.