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(2.3)^x=(0.23)^y=1000 Find 1/x-1/y

2 min read Jun 16, 2024
(2.3)^x=(0.23)^y=1000 Find 1/x-1/y

Solving for 1/x - 1/y in the Equation (2.3)^x = (0.23)^y = 1000

This problem involves working with exponential equations and manipulating them to find a solution. Let's break down the steps to find the value of 1/x - 1/y:

1. Expressing the Equations in a Common Base:

We can express all the numbers in the equations using a common base. Since 2.3 and 0.23 are related, let's use base 2.3:

  • (2.3)^x = 1000
  • (0.23)^y = 1000
  • (2.3)^(-y) = 1000 (Since 0.23 = 2.3^-1)

Now we have:

  • (2.3)^x = 1000
  • (2.3)^(-y) = 1000

2. Equating Exponents:

Since both equations equal 1000, the exponents must be equal:

  • x = -y

3. Finding 1/x - 1/y:

Let's substitute -y for x in the expression 1/x - 1/y:

1/x - 1/y = 1/(-y) - 1/y = -1/y - 1/y = -2/y

Therefore, the value of 1/x - 1/y is -2/y. Note that we cannot determine the exact numerical value without knowing the value of y.

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