Final answer:
To solve the equation log_5(y-5) + 2log_5(4x-1) = 9 for y, we use logarithmic properties to combine terms and then apply the inverse relationship of logs and exponents to isolate y. The final solution for y in terms of x is y = 5^9/(4x - 1)^2 + 5.
Step-by-step explanation:
The student's question involves solving the equation for y in terms of x. The given equation is log5(y-5)+2log5(4x-1)=9. To solve for y, we'll use properties of logarithms and exponents.
First, we apply the property that allows us to convert the multiplication of logs into a single log of the product of their arguments: 2log5(4x - 1) becomes log5((4x - 1)2). Now, our equation looks like this: log5(y - 5) + log5((4x - 1)2) = 9.
Next, we combine the two logarithmic terms using another logarithmic property: log5(A) + log5(B) = log5(AB). This transforms our equation into: log5((y - 5)(4x - 1)2) = 9.
To isolate y, we use the inverse relationship between logs and exponents, converting log base 5 to an exponent: (y - 5)(4x - 1)2 = 59. Finally, we solve for y: y = 59/(4x - 1)2 + 5.