$1.$ Logarithmic functional form and its benefits

Suppose the population regression model, which shows the relationship between the explanatory variables $($x$1$and x$2)$ and the dependent variable y$,$ is given by:

log$($y$)=$$0+$$1$x$1+$$2$x$2+$u

Suppose the model has been estimated, and the results are as follows:

log$($y$)=5\mathrm{.}1+2\mathrm{.}6$x$1+0\mathrm{.}9$x$2$

From the coefficient estimate of $2,$ holding x$1$constant$,$ when x$2$increases by $1$ unit, y$$by$\%,$ which is only an approximation that becomes more inaccurate as the change in log$($y$)$increases$.$

However, when x$2$increases by one unit, y$$by exactly$\%.$ When x$2$decreases by one unit, y$$by exactly$\%.$