This document contains a summary of key concepts and example problems from a math textbook chapter on exponents, logarithms, and exponential and logarithmic functions. It covers graphing and solving exponential equations/inequalities, properties and applications of logarithms including the Richter scale, modeling population growth and decay using exponential and logarithmic functions, and decibel calculations for sound intensity. Example problems are provided to illustrate concepts like compound interest, advertising models, endangered species populations, and muffler noise reductions.

1. MATH CHAPTER 7
Hind Saed Al-Awadi and Shamma Hassan | 1103 |
Mr.Mustafa Ayoubi | math
CCSS PROBLEMS

2. LESSON 7.1

3. 7-1 GRAPHING EXPONENTIAL
FUNCTIONS
• A virus spreads through
a network of computers
such that each minute,
25% more computers
are infected. If the virus
began at only one
computer, graph the
function for the first
hour of the spread of
the virus.
• 𝑌 = 𝑎(𝑏) 𝑥
• A=1, b=1.25
• 𝑦 = 1(1.25) 𝑥

4. LESSON 7.2

5. 7-2 SOLVING EXPONENTIAL EQUATIONS
AND INEQUALITIES
• In 2009, my-lien received $10,000 from her grandmother. Her parents invested all of the money, and by 2021, the
amount will have grown to $16,960.
• A. write an exponential function that could be used to model the money y. write the function in terms of x, the
number of years since 2009.
• 𝑦 = a𝑏 𝑥
• 16960 = 10000(𝑏)12
• 1.045=b
• 𝑦 = 10000(1.045) 𝑥
• B. Assume that the amount of money continues to grow at the same rate. What would be the balance in the
account in 2031?
• 𝑦 = 10000(1.045)22
• Y=$26,336.5

6. LESSON 7.3

7. 7-3 LOGARITHMS AND LOGARITHMIC
FUNCTIONS
• CCSS:modeling, in general, the more money a company spends on advertising, the higher the sales. The
amount of money in sales for a company, in thousands, can be modeled by the equation 𝑆(𝑎) = 10 +
20 log4 𝑎 + 1 where a is the amount of money spent on advertising in thousands, when 𝑎 ≥ 0
• a. The value of S(0) ≈ 10, which means that if $10 is spent on advertising, $10,000 is returned in sales. Find the
values of S(3), S(15), and S(63).
• b. Interpret the meaning of each function value in the context of the problem.
• If $3000 is spent on advertising, $30,000 is returned in sales. If $15,000 is spent on advertising, $50,000 is
returned in sales. If $63,000 is spent on advertising, $70,000 is returned in sales.

8. 7-3
• c. Graph the function.
• The function represents a transformation of the
• graph of .
• a = 20: The graph is expanded vertically.
• h = –1: The graph is translated 1 unit to the left.
• k = 10: The graph is translated 10 units up.
• d. Use the graph in part c and your answers from part a to explain why
the money spent in advertising becomes less “efficient” as it is used in
larger amounts.
• Because eventually the graph plateaus and no matter how much money
you spend you are still returning about the same in sales.

10. 7-4 SOLVING LOGARITHMIC EQUATIONS
AND INEQUALITIES
• CCSS:Modeling. The magnitude of an earthquake is measured on a logarithmic scale called the richter scale. the magnitude M is given 𝑚 =
log10 𝑥,where x represents the amplitude od the seismic wave causing ground motion.
• a. How many times as great is the amplitude caused by an earthquake with a Richter scale rating of 8 as an aftershock with a Richter scale
rating of 5?
• 8=logx
• 100000000=x
• 5=logx
• 100000-x
• 100000000-100000=1000
• A richter scale of 8 is1000 times greater than a richter scaled of 5
• b. In 1906, San Francisco was almost completely destroyed by a 7.8 magnitude earthquake. In 1911, an earthquake estimated at magnitude
8.1 occurred along the New Madrid fault in the Mississippi River Valley. How many times greater was the New Madrid earthquake than the
San Francisco earthquake? New madrid earthquake is 1.99 times bigger than san francisco

11. LESSON 7.5

12. 7-5
• CCSS:Perserverance. The humpback whale is an
endangered species. Suppose there are 5000
humpback whales in existence today, and the
population decreases at a rate of 4% a year.
• a. Write a logarithmic function for the time in years
based upon population.
• 𝑡 = log0.96
𝑝
5000
• b. After how long will the population drop below
1000? Round your answer to the nearest year
• It will take 39 years for the population to drop
below 1000

13. LESSON 7.6

14. 7-6
• 22. Loretta had a new muffler installed on her car. The noise level of the engine dropped from 85 decibels to
73 decibles.
• How many times the minimum intensity of sound detectable by the human ear was the car with the old
muffler , if m is defined to be 1.
• 8.5 = 10 log 1/1
• 8.5 = log 1
• 1 = 108.5
• =316227.76
• The old muffler was about 316 million times louder

15. 7-7
• How many times the minimum intensity of sound detectable by the human ear is the car
with the new muffler? Find the percentage of decrease of the intensity of the sound with
the new muffler?
• 7.3 = 10 log 1/1
• 1 = 107.3
• =19.952
• The muffler is about 20 million times louder
• 19.952/316227.76 = 0.063
• The

16. 7-8
• 40. Five years ago the grizzly bear population in a certain national park
was 325. Today it is 450. Studies Show that the park can support a
population of 750.
• What is the average annual rate of growth in the population if the grizzly
bears reproduce once a year?
• How many years will it take to reach the maximum population if the
population growth continues at the same rate?