W4D1
White Board problems and Answers
Problems
Answers
1.
log7 (7x + 3) = log7 (5x + 9)
2.
log3 (x + 3) + log3 (x − 2) = log3 14
x = 1.3
x = 2.4, −5
3.
log2 (3x + 11) = 5
4.
log(4x − 8) = 3
x = 3.7
x = 4.252
5.
log3 (6x + 3) + log2 8 = log2 128
6.
log4 64 − log4 (x − 12) = log2 8
x = 5.13
x = 6.13
7.
log5 (x + 3) + log5 (x) = log5 (−8 − 3x)
8.
4 log3 (−9x) = 4
x = 7. − 4
x = 8. −
9.
log3
11.
1
81
+ log9 (x + 1) = −5 + log6 36
log4 3x + 2 log3 9 = log4 256
1
3
10.
− 2 log5 (x + 1) = −8
x = 9.0
x = 10.624
12.
log4 (x2 − 6x) = log4 (10 + 3x)
x = 11. 13
x = 12. − 1, 10
Continue Lesson 43: Solve logs Complex
5x+7 = 42x+1
take the log of both sides
log5 5x+7 = log5 42x+1
x + 7 log5 5 = (2x + 1) log5 4
do on a calculator - change the base
x + 7(1) = (2x + 1).86 = 1.72x + .86
.72x = −6.14
x = −8.5
N oSol
log4 (5x + 2) = log2 (5x)
4log4 (5x+2) = 4log2 (5x)
4log4 (5x+2) = 22 log2 (5x)
4log4 (5x+2) = 2log2 (25x
2
)
5x + 2 = 25x2
25x2 − 5x − 2 = 0
(5x + 1)(5x − 2) = 0
1
2
−
x=
5
5
2
x=
5
32x + 4 · 3x − 21 = 0
variable substitution 3x = u
u2 + 4u − 21 = 0
(u + 7)(u − 4) = 0
u = −7
x
3 = −7
log3 (−7) = x
4
x
3 =4
log3 4 = x
x = log3 4 = 1.26
What are Logs Good For 1. Temperature
T = (T0 − TR )e−rt + TR
100 = (212 − 70)e−.046t + 70
30 = 142e−.046t
ln(.2112) = −.046t
−1.55
t=
= 34min
−.046
2. Sound
L = 10 log10 I
a. Whisper 20
b. Rock Concert 120
c. Number Line of loudness (db)
d. 0
e. negative
L
f. I = 10 10
g. La = Lb + 20
3. Richter
r = log10 a
a. Micro Amplitude less than 100
b. Greater than 8
d. A1 = 1000A2
M1 = 3 + M2
Plot the following points on a graph. This is the CDC data on people contracting the flu this fall
Week
Number of Flu Cases
Jan 4
2486
11
1711
10
1639
9
1301
8
830
7
536
6
397
5
312
4
231
3
201
2
145
Oct 16
135
2,400
2,100
1,800
1,500
1,200
900
600
300
1
2
3
4
5
6
7
8
9 10 11 12 13
Figure 1: Cases of Flu in US Oct 16 to Jan 4 103.6(1.3007)x
Can find the equation using two values. Which two points should we use?
First and last - try it. Exponential goes fast so want to use the latest point.
A = P (1 + r)t
135 = P (1 + r)
2486 = P (1 + r)1 2
Divide to get rid of P
18.4 = (1 + r)1 1
1
18.4( ) = 1.3
11
r = .3
135 = P (1.3)
p = 103.6
How many people had the Flu Oct 16? How many people had it 3 months later?
If this were scaled for Kimball and 2 people had the flu. How many would be sick 2 weeks later? 12 weeks later?
What is 10 people started with the flu?
A = 2(1.3)2 = 3.38 = 4people
A = 2(1.3)1 2 = 46.6 = 47people
A = 10(1.3)1 2 = 233people
Richter Activity
one of Richters formulas that relates the magnitude of surface waves of an earthquake, MS, and the energy, E, released in the
quake. This formula is: log E = 11.8 + 1.5MS, where MS is the Richter Scale reading.
1. In 1999, Mexico City was hit by a massive earthquake that measured a magnitude of 7.4 on the Richter Scale. Calculate the
amount of energy released by this earthquake.
2. In 1989, the San Francisco Bay Area in California was hit by an earthquake that measured a magnitude of 7.1 on the Richter
scale. Calculate the amount of energy released by this earthquake.
3. Compare the amount of energy released in the two earthquakes described above.
4. Suppose one earthquake has a magnitude of 2 and another a magnitude of four. What is the ratio of energy released between
the second and first earthquake?
5. Choose two other magnitudes that are in a ratio of 2 and repeat Exercise 4.
6. Write a general expression that gives the ratio of energy released in terms of magnitude when the two magnitudes are in a
ratio 2. What insight does this information g
Richter Activity
Show page 1 with seismogram and nomograph. point out P wave, S wave, amplitude. With this scientists can figure out all sorts
of information about the Quake!
Make PPT with these slides. Get better S-P graph and Nomogram with more minor marks from Virtual
Earthquake File
EX 1:
S-P = 30. A = 23. Need the S-P graph to find the distance. 30 Sec = 290 km. Roughly multiply by 9.7.
Use Nomogram and get M = 5.4 ish Can also use the formula
FORMULA: M = log A + 3 log(8 − b S−P
30 c)(S − P )) − 2.92
M = log A + 3 log(8(S − P )) − 2.92
and
EX
EX
EX
get M = log 23 + 3 log 240 − 2.92 = 5.58
2:
Use page 2 WS? A seismic station is 60 km from the epicenter with a S wave amplitude of 50 mm. 4.4
3:
Magnatude of 6.0 caused a S wave of 20 mm. How far from the epicenter was the seismograph? 510 km
4:
Use the 3 examples of Loma Prieta location and amplitude to find the magnatude.
City
S-P seconds
Distance (km)
S Amplitude
Eureka
49
480
285
Las Vegas
64
600
100
Elko NV
74
690
60
All show Magnitude of 7.1 (discuss different quakes and magnitude)
EX 5:
Local Quakes The table below shows some recent Earthquakes (and not so recent ones) around Tracy. The
distance from Livermore and Magnitude are provided. Use either the formula or a nomogram to determine which Earthquake
was felt most (had the biggest S wave amplitude) here in Livermore.
City
Date
Distance (km)
S-P
Magnitude
S Amp Nom
S Amp Calc
Livermore
Jan 13 2014
0
3.6
1.8
0.4
1.23
Milpitas
Jan 14 2014
38.3
5
2.3
0.5
2.6
Sonoma
Jan 12 2014
116
18
4.5
12
8.8
Clearlake
Jan 21 2014
204.4
18
3.7
1.6
1.4
Eureka
April 1995
494
41
7.2
500
373.6
Coalinga
May 1983
243
21
6.5
500+
555
2.
Plot the following points on a graph. This is the CDC data on people contracting the flu this fall
Week
Number of Flu Cases
Jan 4
2486
11
1711
10
1639
9
1301
8
830
7
536
6
397
5
312
4
231
3
201
2
145
Oct 16
135
2,400
2,100
1,800
1,500
1,200
900
600
300
1
2
3
4
5
6
7
8
9 10 11 12 13
Figure 2: Cases of Flu in US Oct 16 to Jan 4 103.6(1.3007)x
Can find the equation using two values. Which two points should we use?
First and last - try it. Exponential goes fast so want to use the latest point.
A = P (1 + r)t
135 = P (1 + r)
2486 = P (1 + r)1 2
Divide to get rid of P
18.4 = (1 + r)1 1
1
18.4( ) = 1.3
11
r = .3
135 = P (1.3)
p = 103.6
How many people had the Flu Oct 16? How many people had it 3 months later?
If this were scaled for Kimball and 2 people had the flu. How many would be sick 2 weeks later? 12 weeks later?
What is 10 people started with the flu?
A = 2(1.3)2 = 3.38 = 4people
A = 2(1.3)1 2 = 46.6 = 47people
A = 10(1.3)1 2 = 233people
Homework
One of Richters formulas that relates the magnitude of surface waves of an earthquake, MS, and the energy, E, released in the
quake. This formula is: log E = 11.8 + 1.5MS, where MS is the Richter Scale reading.
1. In 1999, Mexico City was hit by a massive earthquake that measured a magnitude of 7.4 on the Richter Scale. Calculate the
amount of energy released by this earthquake.
2. In 1989, the San Francisco Bay Area in California was hit by an earthquake that measured a magnitude of 7.1 on the Richter
scale. Calculate the amount of energy released by this earthquake.
3. Compare the amount of energy released in the two earthquakes described above.
4. Suppose one earthquake has a magnitude of 2 and another a magnitude of four. What is the ratio of energy released between
the second and first earthquake?
5. Choose two other magnitudes that are in a ratio of 2 and repeat problem 4.
6. Write a general expression that gives the ratio of energy released in terms of magnitude when the two magnitudes are in a
ratio 2. What insight does this information give?
Income men vs women
spread of aids (decrease now?)
Exit Pass
1. Are you more comfortable using the formula or the Nomogram?
2. Why do you think that some of the Earthquakes had similar answers using both methods and others were very different?
Income men vs women
spread of aids (decrease now?)