To solve the equation [tex]\( \log_{11}(y+8) + \log_{11} 4 = \log_{11} 60 \)[/tex] using logarithmic properties, follow these steps:
### Step 1: Use the Product Property of Logarithms
The first step is to use the product property of logarithms, which states that [tex]\( \log_b(A) + \log_b(B) = \log_b(A \cdot B) \)[/tex]. For our given equation:
[tex]\[
\log_{11}(y + 8) + \log_{11} 4 = \log_{11} 60
\][/tex]
We can combine the logarithmic terms on the left-hand side:
[tex]\[
\log_{11}((y + 8) \cdot 4) = \log_{11} 60
\][/tex]
### Step 2: Simplify the Expression Inside the Logarithm
Simplify the expression inside the logarithm:
[tex]\[
\log_{11}(4(y + 8)) = \log_{11} 60
\][/tex]
### Step 3: Equate the Arguments of the Logarithms
Since the logarithms on both sides of the equation have the same base and are equal, their arguments must be equal. Therefore, we set the expressions inside the logarithms equal to each other:
[tex]\[
4(y + 8) = 60
\][/tex]
### Step 4: Solve for [tex]\( y \)[/tex]
Now, let's solve for [tex]\( y \)[/tex] step-by-step:
1. Distribute the 4 on the left-hand side:
[tex]\[
4y + 32 = 60
\][/tex]
2. Subtract 32 from both sides of the equation:
[tex]\[
4y = 28
\][/tex]
3. Divide both sides by 4 to isolate [tex]\( y \)[/tex]:
[tex]\[
y = \frac{28}{4}
\][/tex]
4. Simplify the division:
[tex]\[
y = 7
\][/tex]
### Conclusion
The solution to the equation [tex]\( \log_{11}(y + 8) + \log_{11} 4 = \log_{11} 60 \)[/tex] is [tex]\( y = 7 \)[/tex].
