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If a^x = b^y and a = b^k, what is the relationship between x, y, and k?

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Question: If a^x = b^y and a = b^k, what is the relationship between x, y, and k?

Options:

  1. x = ky
  2. y = kx
  3. x + y = k
  4. x - y = k

Correct Answer: x = ky

Solution: 

Substituting a = b^k into a^x = b^y gives (b^k)^x = b^y, leading to kx = y, hence x = ky.

If a^x = b^y and a = b^k, what is the relationship between x, y, and k?

Practice Questions

Q1
If a^x = b^y and a = b^k, what is the relationship between x, y, and k?
  1. x = ky
  2. y = kx
  3. x + y = k
  4. x - y = k

Questions & Step-by-Step Solutions

If a^x = b^y and a = b^k, what is the relationship between x, y, and k?
  • Step 1: Start with the equation a^x = b^y.
  • Step 2: We know that a = b^k. Substitute b^k for a in the equation.
  • Step 3: This gives us (b^k)^x = b^y.
  • Step 4: Use the power of a power rule: (b^k)^x can be rewritten as b^(kx).
  • Step 5: Now we have b^(kx) = b^y.
  • Step 6: Since the bases (b) are the same, we can set the exponents equal to each other: kx = y.
  • Step 7: To find the relationship between x, y, and k, we can rearrange this equation to get x = y/k.
  • Exponential Equations – The question tests the understanding of how to manipulate and solve equations involving exponents.
  • Substitution – It assesses the ability to substitute one expression into another to simplify and solve for variables.
  • Relationships between Variables – The question examines the relationship between the variables x, y, and k derived from the given equations.
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