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# 7.1: Logarithms

Difficulty Level: At Grade Created by: CK-12

Name: __________________

## Introduction to Logarithms

1. On day $0$, you have $1\;\mathrm{penny}$. Every day, you double.
1. How many pennies do you have on day $10$?
2. How many pennies do you have on day $n$?
3. On what day do you have $32\;\mathrm{pennies}$? Before you answer, express this question as an equation, where $x$ is the variable you want to solve for.
4. Now, what is $x$?
2. A radioactive substance is decaying. There is currently $100g$ of the substance.
1. How much substance will there be after $3$ half-lives?
2. How much substance will there be after $n$ half-lives?
3. After how many half-lives will there be $1 g$ of the substance left? Before you answer, express this question as an equation, where $x$ is the variable you want to solve for.
4. Now, what is $x$? (Your answer will be approximate.)

In both of the problems above, part (d) required you to invert the normal exponential function. Instead of going from time to amount, it asked you to go from amount to time. (This is what an inverse function does—it goes the other way—remember?)

So let’s go ahead and talk formally about an inverse exponential function. Remember that an inverse function goes backward. If $f(x)=2^x$ turns a $3$ into an $8$, then $f^{-1}(x)$ must turn an $8$ into a $3$.

So, fill in the following table (on the left) with a bunch of $x$ and $y$ values for the mysterious inverse function of $2^x$. Pick $x-$values that will make for easy $y-$values. See if you can find a few $x-$values that make $y$ be $0$ or negative numbers!

On the right, fill in $x$ and $y$ values for the inverse function of $10^x$.

Inverse of $2^x$ Inverse of $2^x$
$x$ $y$
$8$ $3$
Inverse of $10^x$ Inverse of $10^x$
$x$ $y$

Now, let’s see if we can get a bit of a handle on this type of function.

In some ways, it’s like a square root. $\sqrt{x}$ is the inverse of $x^2$. When you see $\sqrt{x}$ you are really seeing a mathematical question: “What number, squared, gives me $x$?”

Now, we have the inverse of $2^x$ (which is quite different from $x^2$ of course). But this new function is also a question: see if you can figure out what it is. That is, try to write a question that will reliably get me from the left-hand column to the right-hand column in the first table above.

Do the same for the second table above.

Now, come up with a word problem of your own, similar to the first two in this exercise, but related to compound interest.

Name: ___________________________

Homework—Logs

$\log_28 =$ asks the question: “$2$ to what power is $8$?” Based on that, you can answer the following questions:

1. $\log_28 =$

2. $\log_39 =$

3. $\log_{10}10 =$

4. $\log_{10}100 =$

5. $\log_{10}1000 =$

6. $\log_{10}1,000,000 =$

7. Looking at your answers to problems $3-6$, what does the $\log_{10}$ tell you about a number?

8. Multiple choice: which of the following is closest to $\log_{10}500$?

A. $1$

B. $1 \frac{1}{2}$

C. $2$

D. $2 \frac{1}{2}$

E. $3$

9. $\log_{10}1 =$

10. $\log_{10}\frac{1}{10} =$

11. $\log_{10}\frac{1}{100} =$

12. $\log_{10}(0.01) =$

13. $\log_{10}{0} =$

14. $\log_{10}(-1) =$

15. $\log_981 =$

16. $\log_9 \frac{1}{9} =$

17. $\log_93 =$

18. $\log_9 \frac{1}{81} =$

19. $\log_9\frac{1}{3} =$

20. $\log_5(5^4)=$

21. $5^{\log_5 4}=$

OK. When I say, $\sqrt{36}=6$, that’s the same thing as saying $6^2=36$. Why? Because $\sqrt{36}$ asks a question: “What squared equals $36$?” So the equation $\sqrt{36}=6$ is providing an answer: “six squared equals $36$.”

You can look at logs in a similar way. If I say $\log_232=5$ I’m asking a question: “$2$ to what power is $32$?” And I’m answering: “$5$. $2$ to the fifth power is $32$.” So saying $\log_2 32=5$ is the same thing as saying $2^5=32$.

Based on this kind of reasoning, rewrite the following logarithm statements as exponent statements.

22. $\log_28 = 3$

23. $\log_3(\frac{1}{3}) = -1$

24. $\log_x(1) = 0$

25. $\log_ax = y$

Now do the same thing backward: rewrite the following exponent statements as logarithm statements.

26. $4^3 = 64$

27. $8^{-\frac{2}{3}} = \frac{1}{4}$

28. $a^b = c$

Finally...you don’t understand a function until you graph it...

29. a. Draw a graph of $y=\log_2x$.Plot at least $5\;\mathrm{points}$ to draw the graph.

b. What are the domain and range of the graph? What does that tell you about this function?

Name: ___________________________

## Properties of Logarithms

1. $\log_2 (2) =$
2. $\log_2 (2 \times 2) =$
3. $\log_2 (2 \times 2 \times 2) =$
4. $\log_2 (2 \times 2 \times 2 \times 2) =$
5. $\log_2 (2 \times 2 \times 2 \times 2 \times 2) =$
6. Based on numbers $1-5$, finish this sentence in words: when you take $\log_2$ of a number, you find:
7. $\log_2 (8) =$
8. $\log_2 (16) =$
9. $\log_2 (8 \times 16) =$
10. $\log_3 (9) =$
11. $\log_3 (27) =$
12. $\log_3 (9 \times 27) =$
13. Based on numbers $7-12$, write an algebraic generalization about logs.
14. Now, let’s dig more deeply into that one. Rewrite problems $7-9$ so they look like problems $1-5$: that is, so the thing you are taking the log of is written as a power of $2$.
1. $^\#7$:
2. $^\#8$:
3. $^\#9$:
4. Based on this rewriting, can you explain why your generalization from $^\#13$ works?
15. $\log_5 (25) =$
16. $\log_5 (\frac{1}{25}) =$
17. $\log_2 (32) =$
18. $\log_2 (\frac{1}{32}) =$
19. Based on numbers $15-18$, write an algebraic generalization about logs.
20. $\log_3 (81) =$
21. $\log_3 (81 \times 81) =$
22. $\log_3 (81)^2 =$
23. $\log_3 (81 \times 81 \times 81) =$
24. $\log_3 (81)^3 =$
25. $\log_3 (81 \times 81 \times 81 \times 81) =$
26. $\log_3 (81)^4 =$
27. Based on numbers $20-26$ write an algebraic generalization about logs.

Name: ___________________________

Homework—Properties of Logarithms

Memorise these three rules

$&\log_x(ab) = \log_xa + \log_xb\\&\log_x \left (\frac{a}{b} \right ) = \log_xa - \log_xb\\&\log_x(a^b) = b \ \log_x a$

1. In class, we demonstrated the first and third rules above. For instance, for the first rule:

$\log_28 = \log_2(2\times 2 \times 2) & = 3\\\log_216 = \log_2(2\times 2 \times 2 \times 2) & = 4\\\log_2(8\times 16) = \log_2[ ( 2 \times 2 \times 2)(2 \times 2 \times 2 \times 2)] & = 7$

This demonstrates that when you multiply two numbers, their logs add.

Now, you come up with a similar demonstration of the second rule of logs, that shows why when you divide two numbers, their logs subtract.

Now we’re going to practice applying those three rules. Take my word for these two facts. (You don’t have to memorize them, but you will be using them for this homework.)

• $\log_58 = 1.29$
• $\log_560 = 2.54$

Now, use those facts to answer the following questions.

2. $\log_5480 =$

(Hint: $480=8\times 60$. So this is $\log_5(8 \times 60)$. Which rule above helps you rewrite this?)

3. How can you use your calculator to test your answer to $^\#2$? (I’m assuming here that you can’t find $\log_5480$ on your calculator, but you can do exponents.) Run the test—did it work?

4. $\log_5\left ( \frac{2}{15} \right ) =$

5. $\log_5 \left ( \frac{15}{2} \right ) =$

6. $\log_564 =$

7. $\log_5(5)^{23} =$

8. $5(\log_523)=$

Simplify, using the $\log(xy)$ property:

9. $\log_a(x \cdot x \cdot x \cdot x)$

10. $\log_a(x \cdot 1)$

Below bracket are different size

Simplify, using the $\log\left ( \frac{x}{y} \right )$ property:

11. $\log_a \left ( \frac{1}{x} \right )$

12. $\log_a \left ( \frac{x}{1} \right )$

13. $\log_a \left ( \frac{x}{x} \right )$

Simplify, using the $\log(x)^b$ property:

14. $\log_a(x)^4$

15.$\log_a(x)^0$

16. $\log_a(x)^{-1}$

17. a. Draw a graph of $y=\log_{\frac{1}{2}}x$. Plot at least $5\;\mathrm{points}$ to draw the graph.

b. What are the domain and range of the graph? What does that tell you about this function?

Name: ___________________________

Using the Laws of Logarithms

$&\log_x(ab) = \log_x a + \log_xb\\&\log_x \left ( \frac{a}{b} \right ) = \log_xa - \log_xb\\&\log_x(a^b) = b \log_xa$

1. Simplify: $\log_3(x^2) - \log_3(x)$

2. Simplify: $\log_3(9x) - \log_3(x)$

3. Simplify: $\frac{\log(x^2)}{\log(x)}$

4. Solve for $x$.

$\log(2x+5)=\log(8 - x)$

5. Solve for $x$:

$\log(3)+ \log(x+2)=\log(12)$

6. Solve for $x$:

$\ln(x)+ \ln(x-5)=\ln(14)$

7. Solve for $y$ in terms of $x$:

$\log(x)=\log(5y)- \log(3y-7)$

So What Are Logarithms Good For, Anyway?

1. Compound Interest. Andy invests $\1,000$ in a bank that pays out $7\%$ interest, compounded annually. Note that your answers to parts (a) and (c) will be numbers, but your answers to parts (b) and (d) will be formulae.

a. After $3\;\mathrm{years}$, how much money does Andy have?

b. After $t\;\mathrm{years}$, how much money m does Andy have? $m(t) =$

c. After how many years does Andy have exactly $\14,198.57$?

d. After how many years $t$ does Andy have $\m$? $t(m) =$

2. Sound Intensity. Sound is a wave in the air—the loudness of the sound is related to the intensity of the wave. The intensity of a whisper is approximately $100$; the intensity of a normal conversation is approximately $1,000,000$. Assuming that a person starts whispering at time $t=0$, and gradually raises his voice to a normal conversational level by time $t=10$, show a possible graph of the intensity of his voice. (*You can’t get the graph exactly, since you only know the beginning and the end, but show the general shape.)

3. That was pretty complicated, wasn’t it? It’s almost impossible to graph or visualize something going from a hundred to a million: the range is too big.

Fortunately, sound volume is usually not measured in intensity, but in loudness, which are defined by the formula: $L=10 \log_{10}I$, where $L$ is the loudness (measured in decibels), and $I$ is the intensity.

a. What is the loudness, in decibels, of a whisper?

b. What is the loudness, in decibels, of a normal conversation?

c. Now do the graph again—showing an evolution from whisper to conversation in $30\;\mathrm {seconds}$—but this time, graph loudness instead of intensity.

d. That was a heck of a lot nicer, wasn’t it? (This one is sort of rhetorical.)

e. The quietest sound a human being can hear is intensity $1$. What is the loudness of that sound?

f. The sound of a jet engine—which is roughly when things get so loud they are painful—is loudness $120\;\mathrm{decibels}$. What is the intensity of that sound?

g. The formula $I$ gave above gives loudness as a function of intensity. Write the opposite function, that gives intensity as a function of loudness.

h. If sound $A$ is twenty decibels higher than sound $B$, how much more intense is it?

4. Earthquake intensity. When an Earthquake occurs, seismic detectors register the shaking of the ground, and are able to measure the “amplitude” (fancy word for “how big they are”) of the waves. However, just like sound intensity, this amplitude varies so much that it is very difficult to graph or work with. So Earthquakes are measured on the Richter scale which is the $\log_{10}$ of the amplitude $(r=\log_{10}a)$.

a. A “microearthquake” is defined as $2.0$ or less on the Richter scale. Microearthquakes are not felt by people, and are detectable only by local seismic detectors. If $a$ is the amplitude of an earthquake, write an inequality that must be true for it to be classified as a microearthquake.

b. A “great earthquake” has amplitude of $100,000,000$ or more. There is generally one great earthquake somewhere in the world each year. If $r$ is the measurement of an earthquake on the Richter scale, write an inequality that must be true for it to be classified as a great earthquake.

c. Imagine trying to show, on a graph, the amplitudes of a bunch of earthquakes, ranging from microearthquakes to great earthquakes. (Go on, just imagine it—I’m not going to make you do it.) A lot easier with the Richter scale, ain’t it?

d. Two Earthquakes are measured—the second one has $1000$ times the amplitude of the first. What is the difference in their measurements on the Richter scale?

5. $\mathrm{pH}$. In Chemistry, a very important quantity is the concentration of Hydrogen ions, written as $[H^+]$—this is related to the acidity of a liquid. In a normal pond, the concentration of Hydrogen ions is around $10^{-6}\;\mathrm{moles/liter}$. (In other words, every liter of water has about $10^{-6}$, or $\frac{1}{1,000,000}\;\mathrm{moles}$ of Hydrogen ions.) Now, acid rain begins to fall on that pond, and the concentration of Hydrogen ions begins to go up, until the concentration is $10^{-4}\;\mathrm{ moles/liter}$ (every liter has $\frac{1}{10,000}\;\mathrm{moles}$ of $H^+$).

a. How much did the concentration go up by?

b. Acidity is usually not measured as concentration (because the numbers are very unmanageable, as you can see), but as $\mathrm{pH}$, which is defined as $-\log_{10}[H^+]$. What is the $pH$ of the normal pond?

c. What is the $\mathrm{pH}$ of the pond after the acid rain?

6. Based on numbers $2-5$, write a brief description of what kind of function generally leads scientists to want to use a logarithmic scale.

Name: ___________________________

Homework: What Are Logarithms Good For, Anyway?

1. I invest $\300$ in a bank that pays $5\%$ interest, compounded annually. So after $t\;\mathrm{years}$, I have $300(1.05)^t$ dollars in the bank. When I come back, I find that my account is worth $\1000$. How many years has it been? Your answer will not be a number—it will be a formula with a log in it.

2. The $\mathrm{pH}$ of a substance is given by the formula $\mathrm{pH} = -\log_{10}[H^+]$, where $[H^+]$ is the concentration of Hydrogen ions.

a. If the Hydrogen concentration is $\frac{1}{10,000}$, what is the $\mathrm{pH}$?

b. If the Hydrogen concentration is $\frac{1}{1,000,000}$, what is the $\mathrm{pH}$?

c. What happens to the $\mathrm{pH}$ every time the Hydrogen concentration divides by $10$?

You may have noticed that all our logarithmic functions use the base $10$. Because this is so common, it is given a special name: the common log. When you see something like $\log(x)$ with no base written at all, that means the log is $10$. (So $\log(x)$ is a shorthand way of writing $\log_{10}(x)$, just like $\sqrt{x}$ is a shorthand way of writing $\sqrt[2]{x}$. With roots, if you don’t see a little number there, you assume a $2$. With logs, you assume a $10$.)

3. In the space below, write the question that $\log(x)$ asks.

Use the common log to answer the following questions.

4. $\log100$

5. $\log1,000$

6. $\log 10,000$

7. $\log$ ($1$ with $n \ 0s$ after it)

8. $\log 500$ (use the log button on your calculator)

OK, so the $\log$ button on your calculator does common logs, that is, logs base $10$.

There is one other $\log$ button on your calculator. It is called the “natural log,” and it is written $\ln$ (which sort of stands for “natural log” only backward—personally, I blame the French).

$\ln$ means the log to the base $e$. What is $e$? It’s a long ugly number—kind of like $\pi$ only different—it goes on forever and you can only approximate it, but it is somewhere around $2.7$. Answer the following questions about the natural log.

9. $\ln(e) =$

10. $\ln(1) =$

11. $\ln(0) =$

12. $\ln(e^5) =$

13. $\ln(3) =$ (*this is the only one that requires the $\ln$ button on your calculator)

Name: _______________

Sample Test: Logarithms

1. $\log_33 =$

2. $\log_39 =$

3. $\log_327 =$

4. $\log_330 =$

(approximately)

5. $\log_31 =$

6. $\log_3(\frac{1}{3}) =$

7. $\log_3 \left ( \frac{1}{9} \right ) =$

8. $\log_3(-3) =$

9. $\log_93 =$

10 $3^{\log_38} =$

11.$\log_{-3}9 =$

12. $\log 100,000 =$

13. $\log \frac{1}{100,000} =$

14. $\ln e^3 =$

15. $\ln 4 =$

16. Rewrite as a logarithm equation (no exponents): $q^z = p$

17. Rewrite as an exponent equation (no logs): $\log_wg = j$

For questions $18-22$, assume that...

$\log_512 & = 1.544\\\log_520 & = 1.861$

18. $\log_5240 =$

19. $\log_5 \left ( \frac{3}{5} \right ) =$

20. $\log_5 1 \frac{2}{3}=$

21. $\log_5 \left ( \frac{3}{5} \right )^2 =$

22. $\log_5400 =$

23. Graph $y = -\log_2x + 2$.

24. What are the domain and range of the graph you drew in $^\#23$?

25. I invest $\200$ in a bank that pays $4\%$ interest, compounded annually. So after $y\;\mathrm{years}$, I have $200 (1.04)^y$ dollars in the bank. When I come back, I find that my account is worth $\1000$. How many years has it been? Your answer will be a formula with a log in it.

26. The “loudness” of a sound is given by the formula $L = 10 \log I$, where $L$ is the loudness (measured in decibels), and $I$ is the intensity of the sound wave.

a. If the sound wave intensity is $10$, what is the loudness?

b. If the sound wave intensity is $10,000$, what is the loudness?

c. If the sound wave intensity is $10,000,000$, what is the loudness?

d. What happens to the loudness every time the sound wave intensity multiplies by $1,000$?

27. Solve for $x$.

$\ln (3)+ \ln(x) = \ln(21)$

28. solve for $x$.

$\log_2(x)+ \log_2(x+10)= \log_2(11)$

Extra credit: Solve for $x. \ e^x =$(cabin)

Feb 23, 2012

Apr 29, 2014