1. Find the inverse of the logarithmic function f defined by f(y) = 2 log5 (2y-8) + 3.
2. If x > y > 1, what is the largest possible value of logx (x/y) + logy (y/x)?
f = 2log5(2y-8) + 3
f = log5(2y-8)^2 + log5 125
f-3 = log5((2y-8)^2)
5^(f-3) = (2y-8)^2
5^((f-3)/2) = 2y-8
2y = 8 + 5^((f-3)/2)
y = 4 + 1/2 √(5^(f-3))
so, the inverse function is
4 + 1/2 √(5^(y-3))
logx(x/y) = logx(x) - logx(y) = 1 - logx(y)
logy(y/x) = 1 - logy(x)
adding them, we have
2 - logx(y) - logy(x)
2 - logy(x) - 1/logy(x)
we know logy(x) > 1, so if we call that z, we want the maximum value of 2 - z - 1/z
That function achieves a maximum of 0 when z=1. That is, as y->x. So, technically, 0 is the least upper bound, but there is no maximum number less than zero.
1. To find the inverse of a function, we need to switch the roles of x and y and solve for y in terms of x.
Let's start with the given function f(y) = 2 log5 (2y-8) + 3.
Step 1: Replace f(y) with x.
x = 2 log5 (2y-8) + 3
Step 2: Subtract 3 from both sides of the equation.
x - 3 = 2 log5 (2y-8)
Step 3: Divide both sides of the equation by 2 to isolate the logarithmic expression.
(x - 3) / 2 = log5 (2y-8)
Step 4: Rewrite the equation using exponential form.
5^((x - 3) / 2) = 2y - 8
Step 5: Add 8 to both sides of the equation.
5^((x - 3) / 2) + 8 = 2y
Step 6: Divide both sides of the equation by 2 to solve for y.
y = (5^((x - 3) / 2) + 8) / 2
The inverse of the given logarithmic function f(y) is f^(-1)(x) = (5^((x - 3) / 2) + 8) / 2.
2. To find the largest possible value of logx (x/y) + logy (y/x), we need to maximize the sum of the two logarithmic expressions.
Let's analyze the given expression: logx (x/y) + logy (y/x)
Step 1: Use the logarithmic properties to simplify the expression:
logx (x/y) + logy (y/x) = logx (x/y) + logy (1/(x/y))
= logx (x/y) + logy (y/x^(-1))
= logx (x/y) + (-logy (x/y))
= logx (x/y) - logy (x/y)
Step 2: Combine the logarithmic terms using the properties of logarithms:
= logx (x/y * y/x)
= logx (xy / xy)
= logx (1)
= 0
Therefore, the largest possible value of logx (x/y) + logy (y/x) is 0.