How do you solve the equation log4(y) = 9 log4(3) log4(81)?

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To solve the equation log4(y) = 9 log4(3) log4(81), we will follow these steps:

  1. First, simplify log4(81). Since 81 = 34, we can rewrite it as: 
    2. log4(81) = log4(34) = 4 * log4(3)
  2. Now, substituting this back into our equation gives:
    3. log4(y) = 9 * log4(3) * (4 * log4(3))
  3. This simplifies to:
    4. log4(y) = 36 * (log4(3))2
  4. Next, to isolate y, we convert from logarithmic form to exponential form. Using the definition of logarithms: 
    5. y = 4{36 * (log4(3))2}
  5. We can further simplify this since we know that a = log4(3) means 4a = 3. Thus:
    6. y = 336 * a
  6. Substituting back a = log4(3), we focus on the calculations involving the exponent:
    7. y = 336 * log4(3)
  7. Now, we remember that we can express this as a power of 4 to simplify:
    8. y = 436} where we substitute back for 3 if required in terms of 4.
  8. Finally, this provides a complete representation of y.

The solution is thus expressed as a relationship of powers, focusing on expressions in terms of 4 and 3. Note that depending on specific values or further simplification aspects, we may adjust final expressions accordingly.

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