### Brain Teasers

# 2 < 1

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:

Proof:

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

Question:

Bob found the problem almost immediately. Can you?

Proof:

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

Question:

Bob found the problem almost immediately. Can you?

### Hint

Remember that the logarithm of any number less than 1 is less than 0.### Answer

When 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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