ggplot2: using square brackets and superscript in axis title - r

Question

I have a plot where each axis has been log10 transformed. For one of my axis titles I would like to use both a square bracket ([]) and a superscript. How can I do this?
Example Data
library(dplyr)
library(ggplot2)
set.seed(123)
df <- data.frame(matrix(ncol = 2, nrow = 20))
colnames(df)[1:2] <- c('x','y')
df$x <- rnorm(20,1000,100)
df$y <- rnorm(20,1000,100)
df <- df %>%
mutate(log_x = log10(x),
log_y = log10(y))
Here is an example of the figure I am trying to make. I need to know how to make the -2 on the x-axis superscripted.
df %>%
ggplot(aes(x = log_x, y = log_y)) +
geom_point() +
labs(x = expression(log[10]~"[Area (m^-2)]"),
y = expression(log[10]~"[ Time Variable (months)]")) +
theme_bw() +
theme(axis.text.x = element_text(size = 16, color = "black"),
axis.text.y = element_text(size = 16, color = "black"),
axis.title = element_text(size = 16, color = "black"),
panel.grid = element_blank(),
panel.background = element_blank())

Answer 1

#MrFlick response provides the correct answer, see below.
df %>%
ggplot(aes(x = log_x, y = log_y)) +
geom_point() +
labs(x = expression(log[10]~"[Area"~ (m^-2) ~"]"),
y = expression(log[10]~"[ Time Variable (months)]")) +
theme_bw() +
theme(axis.text.x = element_text(size = 16, color = "black"),
axis.text.y = element_text(size = 16, color = "black"),
axis.title = element_text(size = 16, color = "black"),
panel.grid = element_blank(),
panel.background = element_blank())