Log linearising (Gali textbook) In Gali (2015)'s textbook Monetary Policy, Inflation, and the Business Cycle, variables in levels are denoted with capital letters, logged variables with lowercase letters. 

However, when a log-linearisation is performed, say of the household optimality condition, why is the result in logs and not in percentage deviations, (lower case letters with a hat, for instance)?

Is there an intermediate step/reason for why we can go from percentage deviations to logs? Thanks!
$$\frac{x_{t+1}}{x_t} \equiv \frac{x_{t}+Δx_{t+1}}{x_t} = 1+\frac{Δx_{t+1}}{x_t}$$

Further,

$$\ln (1+a) \approx a,$$

an acceptable approximation for $|a| &lt;0.1$ and certainly for $|a| &lt;0.05$.