If the law being challenged has something to do with constitutional interpretation, then yes, to a degree. Many (most?) cases don't involve an interpretation of the constitution, but rather of the meaning of a statute. Many of the same general principles apply here, so I'll lump that in for simplicity's sake.
Because the Constitution is broadly drafted, it doesn't even come close to covering every imaginable situation that the courts will face. Even with the help of contemporaneous records and things like the Judiciary Act of 1789, we can only know so much about what the framers were thinking when they wrote the law. However, that doesn't mean that the court has to resort to completely subjective interpretation. It helps if you think about it as a gauntlet.
In many cases, the Constitution is unequivocal. If the constitution says you get a trial by jury, you get a trial by jury. Those cases get kicked out here while the others continue on.
In the vast majority of those cases where the text doesn't explicitly answer the question, this historical knowledge and context I mentioned is enough to resolve the dispute - that's why a huge portion of the difficult cases arising under the Constitution were resolved relatively early, leaving the courts with precedent to guide them.
Now, when you're interpreting a statute passed by Congress, you have to deal with many of the same issues. Most laws that are at issue are fairly old, and the questions of what Congress meant when the laws were enacted are no less challenging. In these cases, the court has a whole set of canons of construction that it uses to determine how to apply the law. These are relatively straight forward and common sense rules, many of which are listed here:
Statutory interpretation - Wikipedia, the free encyclopedia
A few examples:
These canons can be used to reach a result in pretty much every possible case. While it's certainly possible for two honest judges to come to a different interpretation of a statute/the Constitution using these methods, that's not "judicial activism" or the idea of the Constitution as a living document. The living document theory is that rather than concerning itself with what the framers/legislators meant when they said XYZ, the court should just determine what it thinks is the best policy and go with that. From my perspective, that's an unwarranted and improper intrusion upon the lawmaking power.
Each amendment clarifies or reinterprets the part of the Constitution that it refers to. Nobody is disputing that. The point is that while the founders obviously intended that the people be able to redefine the Constitution via the amendment process, there is very little evidence that they intended that the courts be able to redefine the Constitution via the judicial process.