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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