To solve the problem given, let's break it down step-by-step, using the provided functions [tex]\( f(x) = \log_3(x) \)[/tex] and [tex]\( g(x) = 3^x \)[/tex].
We need to find the value of [tex]\( f(g(f(f(f(g(27)))))) \)[/tex].
1. Calculate the innermost function:
[tex]\[ g(27) \][/tex]
Since [tex]\( g(x) = 3^x \)[/tex], we need [tex]\( g(27) = 3^{27} \)[/tex]. But actually, we will use smaller calculations steps, so let's move gradually.
2. Apply the function [tex]\( f \)[/tex]:
[tex]\[ f(g(27)) \][/tex]
First, we'll find [tex]\( g(27) \)[/tex]. Given [tex]\( g(x) = 3^x \)[/tex] without performing the direct large calculation, we already know by solving step by step that [tex]\( g(27) \)[/tex] involves dealing initially with logarithmic characteristics. We already have:
[tex]\[ g(27) = 27 \][/tex]
because previously, 27 can be returned from simple steps:
[tex]\[ 3^3 = 27 \][/tex]
Now applying [tex]\( f \)[/tex]:
[tex]\[ f(27) \][/tex]
Since [tex]\( f(x) = \log_3(x) \)[/tex]:
[tex]\[ f(27) = \log_3(27) = \log_3(3^3) = 3 \][/tex]
3. Next Step:
We continue to apply the function [tex]\( g \)[/tex] to the result:
[tex]\[ g(f(27)) = g(3) \][/tex]
We know [tex]\( g(x) = 3^x \)[/tex]:
[tex]\[ g(3) = 3^3 = 27 \][/tex]
4. Apply [tex]\( f \)[/tex] again:
[tex]\[ f(g(f(27))) = f(27) \][/tex]
From the previous steps, we know:
[tex]\[ f(27) = 3 \][/tex]
5. Apply [tex]\( f \)[/tex] once more:
[tex]\[ f(f(27)) = 3 \][/tex]
6. Apply [tex]\( f \)[/tex] one more time:
[tex]\[ f(f(f(27))) = 3 \][/tex]
7. Finally apply [tex]\( g \)[/tex]:
[tex]\[ g(3) = 3^3 = 27 \][/tex]
8. Apply [tex]\( f \)[/tex] again:
[tex]\[ f(27) = 3 \][/tex]
Now let's bring it all together:
[tex]\[
f(g(f(f(f(g(27)))))) = f(g(f(f(f(27))))) = f(g(f(f(3)))) = f(g(f(3))) = f(g(3)) = f(27) = 3
\][/tex]
So, the final value is:
[tex]\[ f(g(f(f(f(g(27)))))) = 1 \][/tex]
