# Interpretation of interaction term with a logged dependent variable

I would like to know how to interpret the coefficient on an interaction term as a percentage change when the dependent variable is logged. I provide an example below using the mtcars data set in R where am is binary and wt is continuous.

$\ln(mpg)= \beta_0 + \beta_1\cdot wt + \beta_2\cdot am + \beta_3\cdot wt\cdot am$

In this case, I would interpret the main effect on am to mean that when wt=0, mpg increases by 47.7% when am moves from 0 to 1, where 47.7 = exp(0.38981)-1)*100.

I would like to know if coefficient on the interaction can also be interpreted as a percentage change. For example, exponentiating the coefficient for the interaction yields (exp(-0.15391)-1)*100 = -14.3%. My interpretation is that for a one unit increase in wt, mpg decreases by 14.3% more for am=1 versus am=0.

I've already plotted the interaction effects, but because the y-scale is in logged mpg the substantive interpretation isn't obvious even though the plot illustrates the interaction quite clearly.

data(mtcars)

summary(mtcars[,c("mpg", "wt", "am")])

m1 <- lm(log(mpg) ~ wt + am + wt*am, data=mtcars)

summary(m1)

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  3.72680    0.15145  24.607  < 2e-16 ***
wt          -0.24148    0.03940  -6.129 1.29e-06 ***
am           0.38981    0.21383   1.823   0.0790 .
wt:am       -0.15391    0.07245  -2.124   0.0426 *