Answer:
y = 6
Step-by-step explanation:
Given logarithmic equation:
[tex]\log_{3}(y+5)+\log_{3}(6)=\log_{3}(66)[/tex]
To find the solution to the given logarithmic equation, we can use logarithmic properties.
[tex]\boxed{\begin{array}{c}\underline{\textsf{Logarithmic Properties}}\\\\\textsf{Product law:}\;\;\log_axy=\log_ax + \log_ay\\\\\textsf{Equality law:} \quad \textsf{If $\log_ax=\log_ay$ then $x=y$}\end{array}}[/tex]
Begin by applying the product rule to the left side of the equation:
[tex]\log_{3}((y+5)\cdot 6)=\log_{3}(66)\\\\\\\log_{3}(6y+30)=\log_{3}(66)[/tex]
Next, apply the equality law by equating the arguments:
[tex]6y+30=66[/tex]
Now, we can solve for y:
[tex]6y+30-30=66-30\\\\\\6y=36\\\\\\\dfrac{6y}{6}=\dfrac{36}{6}\\\\\\y=6[/tex]
Therefore, the solution to the given logarithmic equation is:
[tex]\Large\boxed{\boxed{y=9}}[/tex]
