You're a doctor. A patient tells you about a problem with the skin on his back. You're going to examine it, so you say this.
Let's take a look at that.
Make a suggestion using "let's ___" when you're pretty sure that the people you're speaking to will accept your suggestion. If you're not as sure, you can say "why don't we ___":
Guys, why don't we call it a day.
The word "look" is very general. It can mean to look for a long time, a short time, carefully, absent-mindedly, or in many other ways. "Take a look" is more specific. It means to look at something for a short period, usually for a specific purpose.
You can ask someone to take a look at something that needs to be fixed, or for something that's important for them to see. Here's another example from a science classroom. The teacher tells the class to look at something in the textbook:
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