Question Video: Finding the Solution Set of a Logarithmic Equation over the Set of Real Numbers Mathematics • 10th Grade

Find the solution set of logβ (64 logβ π₯) = 4 in β.

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Video Transcript

Find the solution set of log to the base four of 64 log to the base four of π₯ equals four for all real numbers.

We recall that if log base π of π is equal to π, then π to the power of π is equal to π. We will begin this question by letting the expression inside the parentheses be equal to π¦. This means that log to the base four of π¦ is equal to four. This in turn means that π¦ is equal to four to the power of four or four to the fourth power. π¦ is therefore equal to 256. This means that 64 multiplied by log to the base four of π₯ must equal 256.

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