Solving the Equation: 5^(x) = (0.5)^(y) = 1000
In this problem, we are given the equation 5^(x) = (0.5)^(y) = 1000, and we need to find the value of (1)/(x) - (1)/(y).
Step 1: Simplify the Equation
First, let's simplify the equation by rewriting 0.5 as 1/2:
5^(x) = (1/2)^(y) = 1000
Step 2: Convert to Exponential Form
Next, we can convert the equation to exponential form by taking the logarithm base 10 of both sides:
log(5^(x)) = log((1/2)^(y)) = log(1000)
Using the property of logarithms, we can rewrite the equation as:
x log(5) = y log(1/2) = 3 log(10)
Step 3: Solve for x and y
Now, we can solve for x and y separately:
x log(5) = 3 log(10) x = 3 log(10) / log(5)
y log(1/2) = 3 log(10) y = 3 log(10) / log(1/2)
Using the property of logarithms, we can simplify x and y further:
x = 3 / log(5) / log(2) y = 3 / log(1/2) / log(2)
Step 4: Find the Value of (1)/(x) - (1)/(y)
Now that we have expressions for x and y, we can find the value of (1)/(x) - (1)/(y):
(1)/(x) - (1)/(y) = log(2) / 3 - log(2) / 3 = 0
Therefore, the value of (1)/(x) - (1)/(y) is 0.
