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Independent Practice with Common Log Word Problems
1) Use algebraic reasoning with logarithms to solve the following equations for x.
a. log x = 2
b. 15 = 10x
c. 5(10)2x = 60
d. 103x – 1 = 100,000
2) When archaeologists discover remains of an ancient civilization, they use a technique called carbon
dating to estimate the time when the person or animal died or when the artifact was made from
living material.
The amount of radioactive Carbon-14 in such an artifact decreases exponentially according to the
function C(t) = 100(10-0.00005255t), where t is time in years and C(t) is the percent of the Carbon 14
present in the artifact when it was last living material.
a. What is the half-life of Carbon-14, the time when only 50% remains from an original amount?
b. Suppose that a skeleton is discovered that has only 10% of the Carbon-14 that one finds in living
animals. When was that skeleton part of a living animal?
3) The Washington Nationals baseball team was purchased in 2006 for 450 million dollars. If the value
of this investment grows at a rate of 5% compounded yearly, the purchase price of the team in 2006
+ t will be given by V(t) = 450(100.021t).
a. Explain how you can be sure that this function gives the correct value of the investment in 2006.
b. Use the function to estimate the value of the investment in 2010.
c. Use logarithms and other algebraic reasoning to estimate the time when the value of the
investment will be $1 billion ($1,000 million).
4) Suppose that the average rent for a two-bedroom apartment in Indianapolis is currently $750 per
month and increasing at a rate of 8% per year. The function R(t) = 750(100.033t) provides a model for
the pattern of expected increase in monthly rent after t more years.
a. Explain how you know that R(0) = 750.
b. Find the time when the average rent for a two-bedroom apartment will be $1,000 per month, if
inflation continues at the current 8% rate. Show how to find that time in two ways – one by
estimation using a table or graph of R(t) and another using logarithms and algebraic reasoning.
5) If an athlete tries to improve his or her performance by taking an illegal drug, the amount of that
drug in his or her blood will decline exponentially over time, but tests are quite sensitive to small
amounts.
For example, a 200 mg dose of a steroid might decay so that the amount remaining after t days is
given by the rule s(t) = 200(10-0.046t).
a. Explain how you know that s(0) = 200.
b. Estimate the time when only 5 mg of the steroid remains active in the athlete’s blood. Show
how to find that time in two ways – one by estimation using a table or graph of s(t) and another
using logarithms and algebraic reasoning.
6) Suppose that 500 mg of a medicine enters a hospital patient’s bloodstream at noon and decays
exponentially at a rate of 15% per hour.
a. Use the exponential function D(t) = 500(10-0.07t) to predict the amount of medicine active in the
patient’s blood at a time 5 hours later, where t is time in hours.
b. Find the time when only 5% (25 mg) of the original amount of medicine will be active in the
patient’s body. Show how to find that time in two ways – one by estimation by using a table or
graph of D(t) and another using logarithms and algebraic reasoning.