# 1.5 Exponential and logarithmic functions  (Page 9/17)

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[T] The concentration of hydrogen ions in a substance is denoted by $\left[{\text{H}}^{+}\right],$ measured in moles per liter. The pH of a substance is defined by the logarithmic function $\text{pH}=\text{−}\text{log}\left[{\text{H}}^{+}\right].$ This function is used to measure the acidity of a substance. The pH of water is 7. A substance with a pH less than 7 is an acid, whereas one that has a pH of more than 7 is a base.

1. Find the pH of the following substances. Round answers to one digit.
2. Determine whether the substance is an acid or a base.
1. Eggs: $\left[{\text{H}}^{+}\right]=1.6\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-8}$ mol/L
2. Beer: $\left[{\text{H}}^{+}\right]=3.16\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}$ mol/L
3. Tomato Juice: $\left[{\text{H}}^{+}\right]=7.94\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}$ mol/L

i. a. pH = 8 b. Base ii. a. pH = 3 b. Acid iii. a. pH = 4 b. Acid

[T] Iodine-131 is a radioactive substance that decays according to the function $Q\left(t\right)={Q}_{0}·{e}^{-0.08664t},$ where ${Q}_{0}$ is the initial quantity of a sample of the substance and t is in days. Determine how long it takes (to the nearest day) for 95% of a quantity to decay.

[T] According to the World Bank, at the end of 2013 ( $t=0$ ) the U.S. population was 316 million and was increasing according to the following model:

$P\left(t\right)=316{e}^{0.0074t},$

where P is measured in millions of people and t is measured in years after 2013.

1. Based on this model, what will be the population of the United States in 2020?
2. Determine when the U.S. population will be twice what it is in 2013.

a. $~333$ million b. 94 years from 2013, or in 2107

[T] The amount A accumulated after 1000 dollars is invested for t years at an interest rate of 4% is modeled by the function $A\left(t\right)=1000{\left(1.04\right)}^{t}.$

1. Find the amount accumulated after 5 years and 10 years.
2. Determine how long it takes for the original investment to triple.

[T] A bacterial colony grown in a lab is known to double in number in 12 hours. Suppose, initially, there are 1000 bacteria present.

1. Use the exponential function $Q={Q}_{0}{e}^{kt}$ to determine the value $k,$ which is the growth rate of the bacteria. Round to four decimal places.
2. Determine approximately how long it takes for 200,000 bacteria to grow.

a. $k\approx 0.0578$ b. $\approx 92$ hours

[T] The rabbit population on a game reserve doubles every 6 months. Suppose there were 120 rabbits initially.

1. Use the exponential function $P={P}_{0}{a}^{t}$ to determine the growth rate constant $a.$ Round to four decimal places.
2. Use the function in part a. to determine approximately how long it takes for the rabbit population to reach 3500.

[T] The 1906 earthquake in San Francisco had a magnitude of 8.3 on the Richter scale. At the same time, in Japan, an earthquake with magnitude 4.9 caused only minor damage. Approximately how much more energy was released by the San Francisco earthquake than by the Japanese earthquake?

The San Francisco earthquake had ${10}^{3.4}\text{or}\phantom{\rule{0.2em}{0ex}}~2512$ times more energy than the Japan earthquake.

## Chapter review exercises

True or False ? Justify your answer with a proof or a counterexample.

A function is always one-to-one.

$f\circ g=g\circ f,$ assuming f and g are functions.

False

A relation that passes the horizontal and vertical line tests is a one-to-one function.

A relation passing the horizontal line test is a function.

False

For the following problems, state the domain and range of the given functions:

$f={x}^{2}+2x-3,\phantom{\rule{3em}{0ex}}g=\text{ln}\left(x-5\right),\phantom{\rule{3em}{0ex}}h=\frac{1}{x+4}$

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hi guys ....um new here ...integrate my welcome
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