To solve the equation [tex]\( \log(y+8) + \log(114) = \log(1160) \)[/tex] using logarithmic properties, follow these steps:
1. Combine the Logarithms:
Using the property of logarithms that states [tex]\( \log(a) + \log(b) = \log(a \cdot b) \)[/tex], we can combine the two logarithms on the left-hand side of the equation:
[tex]\[
\log(y+8) + \log(114) = \log((y+8) \cdot 114)
\][/tex]
2. Rewrite the Equation:
Substitute this into the original equation to get:
[tex]\[
\log((y+8) \cdot 114) = \log(1160)
\][/tex]
3. Drop the Logarithms:
Since the logarithms on both sides of the equation are equal, their arguments must also be equal. Thus, we can equate the arguments:
[tex]\[
114(y+8) = 1160
\][/tex]
4. Solve for [tex]\( y \)[/tex]:
To isolate [tex]\( y \)[/tex], first divide both sides of the equation by 114:
[tex]\[
y+8 = \frac{1160}{114}
\][/tex]
5. Simplify the Division:
Calculate the value of [tex]\(\frac{1160}{114}\)[/tex]:
[tex]\[
y + 8 = 10.175438596491228
\][/tex]
6. Isolate [tex]\( y \)[/tex]:
Subtract 8 from both sides of the equation to solve for [tex]\( y \)[/tex]:
[tex]\[
y = 10.175438596491228 - 8
\][/tex]
7. Calculate the Value:
Perform the subtraction:
[tex]\[
y = 2.1754385964912277
\][/tex]
Thus, the solution to the equation [tex]\( \log(y+8) + \log(114) = \log(1160) \)[/tex] is [tex]\( y = 2.1754385964912277 \)[/tex]. Additionally, when combined [tex]\( y+8 \)[/tex] equals [tex]\( 10.175438596491228 \)[/tex].
