Extracting lambda value with highest log-likelihood from boxcox output library(MASS)
bc <- boxcox(Volume ~ log(Height) + log(Girth), data = trees)

To find the $\lambda$ value with the highest log-likelihood, this command could be used:
bc$x[which.max(bc$y)]

Is x=log(Height) in this example?
 A: If you only grab the highest lambda from that analysis then you are missing the point of the function and its return.  Part of what it is trying to show you (look at the graph that it produces) is that there is a region of reasonable lambda values.  You should use this and knowledge of the science behind the data to choose a reasonable value.  If the "best" lambda value is 0.43, but the interval includes 0.5 and there is a meaningful reason why a square root transform makes sense then using 0.5 is better in many ways than using 0.43 even though it gives the maximum.
The original paper by Box and Cox talked about understanding the science and finding reasonble candidates for lambda, not just trying everything and using the maximum value even when there is no justification (and there is justification for other values that are almost as good).
A: No, x is not log(Height). If you write bc$x then you are extracting $\lambda$ from the bc object. The bc object gives two values x and y. The x referes to the $\lambda$ (or x axis) and y refers to the value of the log-likelihood. Check out this clip.
A: You can zoom in on box cox if you are trying to find an exact value for lambda. 
MASS::boxcox(fit, lambda = seq(S,F,D))

lambda is defined as a sequence from S to F partitioned by D. D is automatically set to 1/10 you can 'zoom in' by making D smaller, i.e. 1/100. When you zoom in make sure the range from S to F is also shrinking. 
