To solve the equation [tex]\(\log_{11}(y+8) + \log_{11} 4 = \log_{11} 60\)[/tex], we will use properties of logarithms to simplify and solve for [tex]\(y\)[/tex].
### Step-by-Step Solution:
1. Combine the logarithms on the left side using the property of logarithms:
[tex]\[
\log_{11}(a) + \log_{11}(b) = \log_{11}(ab)
\][/tex]
Applying this property, we get:
[tex]\[
\log_{11}(y+8) + \log_{11} 4 = \log_{11}((y+8) \cdot 4)
\][/tex]
2. Rewrite the equation using the combined logarithm:
[tex]\[
\log_{11}(4(y+8)) = \log_{11} 60
\][/tex]
3. Since the logarithms with the same base are equal, their arguments must also be equal:
[tex]\[
4(y + 8) = 60
\][/tex]
4. Solve the resulting equation for [tex]\(y\)[/tex]:
[tex]\[
4y + 32 = 60
\][/tex]
Subtract 32 from both sides:
[tex]\[
4y = 28
\][/tex]
Divide both sides by 4:
[tex]\[
y = 7
\][/tex]
Thus, the solution to the equation [tex]\(\log_{11}(y+8) + \log_{11} 4 = \log_{11} 60\)[/tex] is:
[tex]\[
y = 7
\][/tex]
