Why can't I graph y=ln(sin(x)) or sin(ln(x))?

Question

Avery Alexander · Accepted Answer

It all depends on what package you use!!

Abigail Allen · Answer

undefined You can't graph $ y = \ln(\sin(x)) $ because the natural logarithm function is undefined for negative and zero values, and the sine function oscillates between -1 and 1. Similarly, $ y = \sin(\ln(x)) $ is not graphable for the same reason, as the sine function's input must be within the range $[-1, 1]$.

HIX Tutor · Answer

undefined Great question! Let's think about each part.

1. **What is $y = \ln(\sin(x))$?** 
   - The $\sin(x)$ part must be greater than 0 to use $\ln$. 
   - When does $\sin(x)$ become 0? 
   - It’s 0 at $x = n\pi$ (where $n$ is an integer). 
   - So we can't use $\ln$ when $\sin(x) = 0$.

2. **Now, what about $y = \sin(\ln(x))$?**
   - The $\ln(x)$ part needs $x > 0$. 
   - What happens when $x \leq 0$? 
   - $\ln(x)$ is undefined, so we can't graph it for those values.

3. **What do both equations have in common?**
   - They both have parts where they can’t be used: 
     - For $\ln(\sin(x))$, when $\sin(x) \leq 0$. 
     - For $\sin(\ln(x))$, when $x \leq 0$.
   
Now, do you see why we can't graph these functions everywhere?