1. Solve: log (y-1) = -log(y-9)
2. Find the inverse of the logarithmic function f defined by f(y)= 2log5 (2y-8)+3.
log(y-1)=-log(y-9)
using the property of logarithms,
log(x) = -log(1/x), then
log(y-1)=log(1/(y-9))
take anti-log on both sides,
y-1 = 1/(y-9)
then solve for quadratic in y after cross-multiplication.
1. To solve the equation log (y-1) = -log(y-9), we can use the properties of logarithms.
First, let's rewrite the equation using the properties of logarithms. The property states that log(a) - log(b) = log(a/b).
So, we have log (y-1) + log (y-9) = 0.
Using the property, we can simplify the equation as log [(y-1)(y-9)] = 0.
Now, since log(x) = 0 if and only if x = 1, we have (y-1)(y-9) = 1.
Expanding the equation, we get y^2 - 10y + 9 = 1.
Rearranging terms, we have y^2 - 10y + 8 = 0.
Now, we can factor or use the quadratic formula to solve for y.
Factoring, we have (y-2)(y-8) = 0.
Setting each factor equal to zero, we get y-2 = 0 or y-8 = 0.
Solving each equation, we find y = 2 or y = 8.
Therefore, the solution to the equation log (y-1) = -log(y-9) is y = 2 or y = 8.
2. To find the inverse of the logarithmic function f defined by f(y) = 2log5 (2y-8) + 3, we need to swap the roles of x and y and solve for y.
Let's start by writing the function in terms of x instead of y.
f(y) = 2log5 (2y-8) + 3
To find the inverse, we will interchange the x and y variables:
x = 2log5 (2y-8) + 3
Now, we can solve for y. We'll follow the steps below:
Step 1: Subtract 3 from both sides:
x - 3 = 2log5 (2y-8)
Step 2: Divide both sides by 2:
(x - 3) / 2 = log5 (2y-8)
Step 3: Rewrite the equation in exponential form:
5^((x - 3) / 2) = 2y - 8
Step 4: Add 8 to both sides:
5^((x - 3) / 2) + 8 = 2y
Step 5: Divide both sides by 2:
(5^((x - 3) / 2) + 8) / 2 = y
Thus, the inverse of the logarithmic function f is given by:
f^(-1)(x) = (5^((x - 3) / 2) + 8) / 2