First, let’s rewrite the given equation for clarity:
[tex]\[ 8 \log_2(9y - 2) + 18 = 50 \][/tex]
### Step 1: Isolate the logarithmic term
Subtract 18 from both sides of the equation to isolate the logarithmic expression:
[tex]\[ 8 \log_2(9y - 2) + 18 - 18 = 50 - 18 \][/tex]
[tex]\[ 8 \log_2(9y - 2) = 32 \][/tex]
### Step 2: Solve for the logarithm
Divide both sides by 8 to simplify further:
[tex]\[ \log_2(9y - 2) = 4 \][/tex]
### Step 3: Rewrite the logarithmic equation in exponential form
Recall the definition of a logarithm: [tex]\(\log_b(a) = c\)[/tex] means [tex]\(b^c = a\)[/tex]. Therefore, we can rewrite the equation as:
[tex]\[ 2^4 = 9y - 2 \][/tex]
[tex]\[ 16 = 9y - 2 \][/tex]
### Step 4: Solve for [tex]\(y\)[/tex]
Add 2 to both sides to isolate [tex]\(9y\)[/tex]:
[tex]\[ 16 + 2 = 9y \][/tex]
[tex]\[ 18 = 9y \][/tex]
Divide both sides by 9 to solve for [tex]\(y\)[/tex]:
[tex]\[ y = \frac{18}{9} \][/tex]
[tex]\[ y = 2 \][/tex]
### Solution Set
Therefore, the exact solution set is [tex]\(\{2\}\)[/tex].
[tex]\[
\boxed{\{2\}}
\][/tex]
