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Find the solution set of log₄ (64 log₄ 𝑥) = 4 in ℝ.

Find the solution set of log to the base four of 64 log to the base four of 𝑥 equals four for all real numbers.

We recall that if log base 𝑎 of 𝑏 is equal to 𝑐, then 𝑎 to the power of 𝑐 is equal to 𝑏. We will begin this question by letting the expression inside the parentheses be equal to 𝑦. This means that log to the base four of 𝑦 is equal to four. This in turn means that 𝑦 is equal to four to the power of four or four to the fourth power. 𝑦 is therefore equal to 256. This means that 64 multiplied by log to the base four of 𝑥 must equal 256.

Dividing both sides of this equation by 64, we have log to the base four of 𝑥 is equal to four. We notice that this is the same equation as we had earlier, except this time the variable is 𝑥 and not 𝑦. 𝑥 is therefore equal to four to the power of four, which is 256. The solution set contains the single real value 256.

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