2 < 1Math brain teasers require computations to solve.
Sue was disappointed after Bob showed her that 2 was not, in fact, equal to 1. She was not quite done, though. She told Bob that while 2 may not be equal to 1, she could show conclusively that 2 was in fact less than 1. Bob said it couldn't be done, of course, and Sue offered the following:
1 < 2
Multiply the equation by log (0.5)
1 * log(0.5) < 2 * log(0.5)
Make the log factors exponents
log(0.5 ^ 1) < log(0.5 ^ 2)
Carry out the exponentiation
log(0.5) < log(0.25)
Raise 10 to the power of each side of the inequality
10^(log(0.5)) < 10^(log(0.25))
By definition of logarithms, 10^log(a) = a, so...
0.5 < 0.25
Then just multiply both sides by 4
2 < 1
Bob found the problem almost immediately. Can you?
HintRemember that the logarithm of any number less than 1 is less than 0.
AnswerWhen Sue multiplied the inequality by log(0.5), she was multiplying by a negative number, and multiplying an inequality by a negative number switches the direction. This first step should result in:
1 * log(0.5) > 2 * log(0.5), which leads to 2 > 1, as expected.
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