Find the error in the following 'proof' that 2

Question

Find the error in the following 'proof' that 2 < 1.

Accepted Answer

Hello, everyone. We have been asked to spot the mistake in the given proof for three is less than one. We have four answer choices which we will evaluate. After we look at our steps should be clear. There is a mistake as three is not less than one. So our first step says one 64th is less than 1/4. And it also states that that is a true statement. It is a true statement. So that step is right. So I'm gonna put a check mark next, we rewrote one 64th in exponential form. So it is 1/4 cubed is less than 1/4. This is true. So that step is right. I'll put a check mark. Next, we took the log of both sides. So we have the log of 1/4 cubed is less than the log of 1/4. So far that still holds up next. We applied the logarithmic rule which means we took the exponent off of the 1/4 and put it as a coefficient. So we have three log, 1/4 is less than the log of 1/4. So far that works. Next, we divided both sides by the log of 1/4. This is where we resulted in three is less than one. So this is where the error is. The error comes from. The fact that the log of 1/4 when we divide both sides by it, we are actually dividing both sides by a negative number because the log of 1/4 is negative. Recall that when you divide an inequality by a negative, it flips the inequality symbol. So we should have that three is greater than one at that step. So now that we've identified our error, let's check our answer choices. So our four answer choices, the first one A says that one 64th is greater than 1/4. That is not it. B the log of 1/4 cubed should be equal to the log of 3/4. That is not the error we identified either C the inequality should be reversed when dividing by the log of 1/4 because the log of 1/4 is less than zero so far, this sounds right. But let's double check the last one D, the inequality should be changed to equality when dividing by the log of 1/4. Because log 1/4 is less than zero, it shouldn't change to inequality. So our answer is that the error is that the inequality should be reversed when dividing by the log of 1/4. So it is answer choice C have a nice day.

Find the error in the following 'proof' that 2 < 1.

Key Concepts

Inequalities

Proof Techniques

Logical Fallacies

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