### Instruction: Determine the value of the expression
\[\log_2 (27 + \log_2 (27 + \log_2 (27 + \cdots))),\]assuming it is positive.### Response: I notice that the expression is a repeated pattern of adding 27 and taking the logarithm base 2.
I wonder what happens if I replace the expression with a variable, say x, and then write an equation relating x to itself using the pattern.
I get \[x = \log_2 (27 + x).\]
This is a transcendental equation, so it might be hard to solve algebraically, but I can try to graph both sides and see where they intersect.
The left-hand side is just a line y = x, and the right-hand side is a logarithmic curve that starts at \(\log_2 27\) and increases slowly as x increases.
I can use a graphing calculator or an online tool to plot both functions and zoom in on the intersection point.
I see that they intersect at approximately x = 5.857, so this is the value of the expression, rounded to three decimal places.
I can check my answer by plugging it back into the original expression and seeing if it makes sense.
I get \[\log_2 (27 + \log_2 (27 + \log_2 (27 + \cdots))) \approx \log_2 (27 + 5.857) \approx 5.857,\]which is consistent with my answer.