set.seed(123)

## Non-linear mean with constant SD
logx <- seq(0.001,1, length.out = 1000)
noise <- rnorm(1000)
y <- logx + noise/4
x <- 10^logx

par(mfrow = c(2,2))
plot(x, y,
     xlab = "x",
     ylab = "y",
     col = "grey")
lines(x, logx)

## => transform the covariate
plot(logx, y,
     xlab = expression(paste(log[10], "(x)")),
     ylab = "y",
     col = "grey")
lines(logx, logx)

## Non-linear mean and non-constant SD
set.seed(124)
x <- seq(1.002, 3, length.out = 1000)
noise <- rnorm(1000)
logy <- -(x + noise/9)
y <- 10^logy

plot(x, y,
     xlab = "x",
     ylab = "y",
     col = "grey")
lines(x, 10^-x)

## => transform the outcome
plot(x, logy,
     xlab = "x",
     ylab = expression(paste(log[10], "(y)")),
     col = "grey")
lines(x, -x)