Chapter 3

Given that a quantity \(Q(t)\) is described by the exponential growth function $$ Q(t)=400 e^{\mathrm{a} .05 t} $$ where \(t\) is measured in minutes, answer the following questions: a. What is the growth constant? b. What quantity is present initially? c. Complete the following table of values:

Express each equation in logarithmic form. $$2^{6}=64$$

Evaluate the expression. a. \(4^{-3} \cdot 4^{5}\) b. \(3^{-3}+3^{6}\)

Given that a quantity \(Q(t)\) exhibiting exponential decay is described by the function $$ Q(t)=2000 e^{-0.06 \mathrm{~s}} $$ where \(t\) is measured in years, answer the following questions: a. What is the decay constant? b. What quantity is present initially? c. Complete the following table of values:

Express each equation in logarithmic form. $$3^{5}=243$$

Evaluate the expression. a. \(\left(2^{-1}\right)^{3}\) b. \(\left(3^{-2}\right)^{3}\)

The growth rate of Escherichia coli, a common bacterium found in the human intestine, is proportional to its size. Under ideal laboratory conditions, when this bacterium is grown in a nutrient broth medium, the number of cells in a culture doubles approximately every \(20 \mathrm{~min}\). a. If the initial cell population is 100 , determine the function \(Q(t)\) that expresses the exponential growth of the number of cells of this bacterium as a function of time \(t\) (in minutes). b. How long will it take for a colony of 100 cells to increase to a population of 1 million? \(\mathbf{c}\), If the initial cell population were 1000 , how would this alter our model?

Express each equation in logarithmic form. $$3^{-2}=\frac{1}{9}$$

Evaluate the expression. a. \(9(9)^{-1 / 2}\) b. \(5(5)^{-1 / 2}\)

The world population at the beginning of 1990 was \(5.3\) billion. Assume that the population continues to grow at the rate of approximately \(2 \% / y e a r\) and find the function \(Q(t)\) that expresses the world population (in billions) as a function of time \(t\) (in years), with \(t=0\) corresponding to the beginning of 1990 . Using this function, complete the following table of values and sketch the graph of the function \(Q\).

Express each equation in logarithmic form. $$5^{-3}=\frac{1}{125}$$

Evaluate the expression. a. \(\left[\left(-\frac{1}{2}\right)^{3}\right]^{-2}\) b. \(\left[\left(-\frac{1}{3}\right)^{2}\right]^{-3}\)

Refer to Exercise \(4 .\) a. If the world population continues to grow at the rate of approximately \(2 \% / y e a r\), find the length of time \(t_{0}\) required for the world population to triple in size. b. Using the time \(t_{0}\) found in part (a), what would be the world population if the growth rate were reduced to \(1.8 \% / y e a r ?\)

Express each equation in logarithmic form. $$\left(\frac{1}{3}\right)^{1}=\frac{1}{3}$$

Evaluate the expression. a. \(\frac{(-3)^{4}(-3)^{5}}{(-3)^{8}}\) b. \(\frac{\left(2^{-4}\right)\left(2^{6}\right)}{2^{-1}}\)

A certain piece of machinery was purchased 3 yr ago by Garland Mills for \(\$ 500,000\). Its present resale value is \(\$ 320,000\). Assuming that the machine's resale value decreases exponentially, what will it be 4 yr from now?

Express each equation in logarithmic form. $$\left(\frac{1}{2}\right)^{-4}=16$$

Evaluate the expression. a. \(3^{1 / 4} \cdot 9^{-5 / 8}\) b. \(2^{3 / 4} \cdot 4^{-3 / 2}\)

If the temperature is constant, then the atmospheric pressure \(P\) (in pounds/square inch) varies with the altitude above sea level \(h\) in accordance with the law $$ P=p_{0} e^{-k h} $$ where \(p_{0}\) is the atmospheric pressure at sea level and \(k\) is a constant. If the atmospheric pressure is \(15 \mathrm{lb} / \mathrm{in} .^{2}\) at sea level and \(12.5 \mathrm{lb} / \mathrm{in} .^{2}\) at \(4000 \mathrm{ft}\), find the atmospheric pressure at an altitude of \(12,000 \mathrm{ft}\).

Express each equation in logarithmic form. $$32^{3 / 5}=8$$

Simplify the expression. a. \(\left(64 x^{9}\right)^{1 / 3}\) b. \(\left(25 x^{3} y^{4}\right)^{1 / 2}\)

Express each equation in logarithmic form. $$81^{3 / 4}=27$$

Simplify the expression. a. \(\left(2 x^{3}\right)\left(-4 x^{-2}\right)\) b. \(\left(4 x^{-2}\right)\left(-3 x^{5}\right)\)

Phosphorus 32 (P-32) has a half-life of \(14.2\) days. If \(100 \mathrm{~g}\) of this substance are present initially, find the amount present after \(t\) days. What amount will be left after \(7.1\) days?

Express each equation in logarithmic form. $$10^{-3}=0.001$$

Simplify the expression. a. \(\frac{6 a^{-5}}{3 a^{-3}}\) b. \(\frac{4 b^{-4}}{12 b^{-6}}\)

Strontium \(90(\mathrm{Sr}-90)\), a radioactive isotope of strontium, is present in the fallout resulting from nuclear explosions. It is especially hazardous to animal life, including humans, because, upon ingestion of contaminated food, it is absorbed into the bone structure. Its half-life is \(27 \mathrm{yr}\). If the amount of \(\mathrm{Sr}-90\) in a certain area is found to be four times the "safe" level, find how much time must elapse before an "acceptable level" is reached.

Express each equation in logarithmic form. $$16^{-1 / 4}=0.5$$

Simplify the expression. a. \(y^{-3 / 2} y^{5 / 3}\) b. \(x^{-3 / 5} x^{8 / 3}\)

Given that \(\log 3 \approx 0.4771\) and \(\log 4 \approx\) 0.6021, find the value of each logarithm. $$\log 12$$

Simplify the expression. a. \(\left(2 x^{3} y^{2}\right)^{3}\) b. \(\left(4 x^{2} y^{2} z^{3}\right)^{2}\)

Given that \(\log 3 \approx 0.4771\) and \(\log 4 \approx\) 0.6021, find the value of each logarithm. $$\log \frac{3}{4}$$

Simplify the expression. a. \(\frac{5^{0}}{\left(2^{-3} x^{-3} y^{2}\right)^{2}}\) b. \(\frac{(x+y)(x-y)}{(x-y)^{0}}\)

The American Court Reporting Institute finds that the average student taking Advanced Machine Shorthand, an intensive 20 -wk course, progresses according to the function $$ Q(t)=120\left(1-e^{-0.05 t}\right)+60 \quad(0 \leq t \leq 20) $$ where \(Q(t)\) measures the number of words (per minute) of dictation that the student can take in machine shorthand after \(t\) wk in the course. Sketch the graph of the function \(Q\) and answer the following questions: a. What is the beginning shorthand speed for the average student in this course? b. What shorthand speed does the average student attain halfway through the course? c. How many words per minute can the average student take after completing this course?

Given that \(\log 3 \approx 0.4771\) and \(\log 4 \approx\) 0.6021, find the value of each logarithm. $$\log 16$$

Solve the equation for \(x\). $$6^{2 x}=6^{4}$$

Based on data compiled by WHO, the number of people living with HIV (human immunodeficiency virus) worldwide from 1985 through 2006 is estimated to be $$ N(t)=\frac{39.88}{1+18.94 e^{-0.2957}} \quad(0 \leq t \leq 21) $$ where \(N(t)\) is measured in millions and \(t\) in years, with \(t=0\) corresponding to the beginning of 1985 . a. How many people were living with HIV worldwide at the beginning of 1985 ? At the beginning of 2005 ? b. Assuming that the trend continued, how many people were living with HIV worldwide at the beginning of \(2008 ?\)

Given that \(\log 3 \approx 0.4771\) and \(\log 4 \approx\) 0.6021, find the value of each logarithm. $$\log \sqrt{3}$$

Solve the equation for \(x\). $$5^{-x}=5^{3}$$

According to data obtained from the CBO, the total federal debt (in trillions of dollars) from 2001 through 2006 is given by $$ f(t)=5.37 e^{0.07 s t} \quad(1 \leq t \leq 6) $$ where \(t\) is measured in years, with \(t=1\) corresponding to 2001\. What was the total federal debt in 2001 ? In 2006 ?

Solve the equation for \(x\). $$3^{3 x-4}=3^{5}$$

Metro Department Store found that \(t\) wk after the end of a sales promotion the volume of sales was given by $$ S(t)=B+A e^{-\lambda t} \quad(0 \leq t \leq 4) $$ where \(B=50,000\) and is equal to the average weekly volume of sales before the promotion. The sales volumes at the end of the first and third weeks were \(\$ 83,515\) and \(\$ 65,055\), respectively. Assume that the sales volume is decreasing exponentially. a. Find the decay constant \(k\). b. Find the sales volume at the end of the fourth week.

Given that \(\log 3 \approx 0.4771\) and \(\log 4 \approx\) 0.6021, find the value of each logarithm. $$\log \frac{1}{300}$$

Solve the equation for \(x\). $$10^{2 x-1}=10^{x+3}$$

Universal Instruments found that the monthly demand for its new line of Galaxy Home Computers \(t\) mo after placing the line on the market was given by $$ D(t)=2000-1500 e^{-0.05 t} \quad(t>0) $$ Graph this function and answer the following questions: a. What is the demand after 1 mo? After 1 yr? After 2 yr? After \(5 \mathrm{yr}\) ? b. At what level is the demand expected to stabilize?

Write the expression as the logarithm of a single quantity. $$2 \ln a+3 \ln b$$

Solve the equation for \(x\). $$(2.1)^{x+2}=(2.1)^{5}$$

Write the expression as the logarithm of a single quantity. $$\frac{1}{2} \ln x+2 \ln y-3 \ln z$$

Solve the equation for \(x\). $$(-1.3)^{x-2}=(-1.3)^{2 x+1}$$

The length (in centimeters) of a typical Pacific halibut \(t\) yr old is approximately $$ f(t)=200\left(1-0.956 e^{-0.18 t}\right) $$ What is the length of a typical 5 -yr-old Pacific halibut?