Question

Using logarithmic properties, what is the solution to log4(y - 9) + log43 = log481? Show all necessary steps.

Answer 1

y = 36

Explanation:

Given logarithmic equation:

`\log_4(y-9)+\log_43=\log_481`

`\textsf{Apply the \textbf{log product law}}: \quad \log_ax + \log_ay=\log_axy`

`\implies \log_4\left(3(y-9)\right)=\log_481`

`\textsf{Apply the \textbf{log equality law}}: \quad \textsf{If $\log_ax=\log_ay$ then $x=y$}`

`\implies 3(y-9)=81`

Divide both sides of the equation by 3:

`\implies (3(y-9))/(3)=(81)/(3)`

`\implies y-9=27`

Add 9 to both sides of the equation:

`\implies y-9+9=27+9`

`\implies y=36`

Therefore, the solution to the given logarithmic equation is y = 36.