Learning Objectives
Problem 1 of 3
Many people mistakenly believe that carbon dating can be used to determine the age of dinosaur bones. In fact, carbon-14 has a half life of about 5730 years which only makes it useful for dating events within the last 35 000 to 50 000 years. However, this half-life does make it useful to archeologists.
If the mammoth bones discovered in this article are found to have only 10% of the amount of uranium-233 (half-life 162 000 years) a live mammoth should have, how long ago did this mammoth die?
Solution
Problem 2 of 3
Solve algebraically: log(3-x) + log(4-3x) - log(x) = log(7)
Solution
log(3-x) + log(4-3x) - log(x) = log(7)
log((3-x)(4-3x))/x = log(7)
log(12-9x-4x+3x^2)/x = log(7)
log(3x^2-13x+12)/x = log(7)
(3x^2-13x+12)/x = 7
3x^2-13x+12 = 7x
3x^2-20x+12 = 0
(3x-2)(x-6) = 0
x = 2/3, 6
Problem 3 of 3
Solve each of the following for x:
(a) 2^{x + 3} = 17^{x}
(b) 8^{log2x} - 25^{log5x} = 4x - 4
Solution
a) = log2^(x+3) = log17^(x)
= (x+3)log2 = (x)log17
= xlog2 + 3log2 = xlog17
= 3log2 = xlog17 - xlog2
= 3log2 / ( log17 - log2 ) = x( log17 - log2 ) / ( log17 - log2 )
= x = 0.9716716133
= x = 0.9717
b) = 2^3log2x - 5^2log5x = 4x-4
= 3x - 2x = 4x - 4
= 3x - 6x = -4
= -3x = -4
= x = 4/3
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