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Why is the graph |loge(x)| reflected in the x-axis for all negative y values and loge|x| is reflecting the whole log graph in the y-axis?

Thanks


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As you know, the modulus function makes negative things positive.

So when you put the whole function in the modulus function, the negative part of the graph becomes positive - hence |loge(x)| only reflects the negative y-values.

But when you put modulus around the x only (i.e. loge|x|), there are two steps involved:
1. First, think about |x| - if you had to list out the numbers it produced, it would look something like: 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5
2. Second, think about what you are doing when you say loge(x). You are plugging in -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 into loge(x). However, because you are saying loge|x|, the -5, -4, -3, -2, -1 are never plugged in in the first place. Instead, you are plugging in 5, 4, 3, 2, 1 (in place of -5, -4, -3, -2, -1). This is why the LHS of loge|x| looks like a reflection of the graph in the y-axis. It is because you are drawing the positive side (RHS) out twice.


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If I'm drawing the positive side out twice wouldn't I get the exact same graph on the same exact position as the other graph? Why must it be reflected in the y axis?
It is reflected because the LHS is drawn "backwards" as you count downwards (5, 4, 3, 2, 1) instead of upwards (1, 2, 3, 4, 5) on the RHS.