1 Exponential form for logarithms (Section 4.2 , page 445)
2 Properties of logarithms (Section 4.2 , page 446)
3 Rules for logarithms (Section 4.3 , page 471)
A logarithmic scale is a scale in which logarithms are used in the measurement of quantities. Suppose we have a quantity that has a small positive range of variation (for example, from 0.00001=10−5
In Example 2 we will see how a small difference on a logarithmic scale results in an impressive increase in the acidity of acid rain compared to an ordinary rain.
1 Define pH.
In chemistry, acidity of a substance is related to the presence of positively charged hydrogen ions. pH represents the effective concentration (activity) of hydrogen ions H+
The pH value is a measure of the acidity or alkalinity of a solution. The pH value for pure water at 25°C (77°F)
Because pH is defined as −log[H+],
Calculate to the nearest tenth the pH value of grapefruit juice if [H+]
Find the hydrogen concentration [H+]
pH=−log[H+]Definition of pH=−log(6.32×10−4)Given [H+]=6.32×10−4=−(log 6.32+log 10−4)Product rule=−log 6.32+4−log 10−4=−(−4)log 10=4≈−0.8007+4Use a calculator.=3.1993≈3.2Simplify.
4.82=−log[H+]Replace pH with 4.82.log[H+]=−4.82Rewrite equation.[H+]=10−4.82Exponential form=100.18×10−5−4.82=−5+0.18≈1.51×10−5Use a calculator.
The [H+]
Find [H+]
How much more acidic is acid rain with a pH value of 3 than an ordinary rain with a pH value of 6?
We have
so that
Similarly,
Therefore,
So the hydrogen ion concentration in this acid rain is 1000 times greater than that in ordinary rain. That is, this acid rain is 1000 times more acidic than the ordinary rain.
How much more acidic is an acid rain with a pH value of 2.8 than an ordinary rain with a pH value of 6.2?
Table 4.8 lists a few typical pH values.
Solution | pH Value |
---|---|
Battery Acid | 1 |
Lemon Juice | 2 |
Stomach Acid | 2–3 |
Vinegar | 3 |
Milk | 6–7 |
Baking Soda, Seawater | 8–9 |
Milk of Magnesia | 9–10 |
Ammonia | 10–11 |
Drain opener | 10–12 |
Lye | 13 |
2 Define the Richter scale for measuring earthquake intensity.
An earthquake is the vibration, sometimes violent, of Earth’s surface that follows a release of energy in Earth’s crust. Earth’s crust is composed of about 20 huge plates that float on the molten material beneath the crust. These plates slowly move over, under, and past each other. Sometimes the movement is gradual. At other times, the plates are locked together, unable to release the accumulating energy. When this energy grows strong enough, the plates break free and vibrations called “seismic waves” or earthquakes are generated.
The Richter scale was invented in the 1930s by the American scientist Dr. Charles Richter to measure the magnitude of an earthquake. It is based on the idea of comparing the intensity of an earthquake with that of a zero-level earthquake. He defined the zero-level earthquake as an earthquake whose seismographic reading measures 1 micron (1000 microns=1 millimeter,
Let us denote the intensity of a zero-level earthquake by I0.
The magnitude of an earthquake is 4.0 on the Richter scale. What is the intensity of this earthquake?
The intensity of this earthquake is 10,000 times the intensity of I0
The magnitude of an earthquake is 6.5 on the Richter scale. What is the intensity of this earthquake?
Compare the intensity of the Mexico City earthquake of 1985, which registered 8.1 on the Richter scale, to that of the San Francisco area earthquake of 1989, which measured 6.9 on the Richter scale.
Let IM
Divide IM
This equation shows that the intensity of the Mexico City earthquake was about 16 times that of the San Francisco area earthquake.
The magnitudes of the earthquakes of Mozambique (2006) and Southern California (2005) were 7.0 and 5.2, respectively. Compare their intensities.
The magnitude M of an earthquake is related to its released energy E (measured in joules) and is approximated by the equation
We can rewrite this equation in the exponential form as
or
The energy E0
Date | Location | Magnitude |
---|---|---|
April 1906 | San Francisco | 7.8 |
December 1908 | Messina, Italy | 7.5 |
December 1920 | Gansu, China | 8.6 |
September 1923 | Sagami Bay, Japan | 8.3 |
February 1931 | New Zealand | 7.9 |
August 1950 | Assam, India | 8.7 |
July 1976 | Tangshan, China | 8.0 |
September 1985 | Mexico City | 8.1 |
June 1990 | Iran | 7.7 |
May 1997 | Iran | 7.5 |
December 2004 | Sumatra, Indonesia | 9.0 |
March 2005 | Sumatra, Indonesia | 8.6 |
February 2010 | Maule, Chile | 8.8 |
March 2011 | Tohoku, Japan | 9.0 |
Source: National Earthquake Information Center, U.S. Geological Survey. |
If two earthquakes have magnitude M1
Compare the estimated energies released by the San Francisco earthquake of 1906 and the Northridge earthquake of 1994 with magnitude of 6.7.
Let M1906
With M1906=7.8
The 1906 earthquake released more than 45 times as much energy as the 1994 earthquake.
Compare the energies released by the Japan earthquake of 2011 and Iran earthquake of 1997 (see Table 4.9 ).
3 Define scales for measuring sound.
The intensity of a sound wave is defined as the amount of power the wave transmits through a given area. The intensity of a sound is measured in watts per square meters (abbreviated W /m2
Because the range of intensities that the human ear can detect is so large, a logarithmic scale called decibels (abbreviated dB) is used to measure the loudness of a sound. A decibel (as the name suggests), is one-tenth of a bel, a unit named after the telephone inventor Alexander Graham Bell.
If we substitute I=I0
So the threshold of hearing has 0 dB loudness.
Table 4.10 lists approximate loudness of some common sounds.
Source | Intensity in W/m2 |
Loudness in dB |
---|---|---|
Threshold of hearing | 10−12 |
0 |
Rustling leaves | 10−11 |
10 |
Whisper | 10−10 |
20 |
Background noise in average home | 10−8 |
40 |
Normal conversation | 10−6 |
60 |
Busy street traffic | 10−5 |
70 |
Vacuum cleaner | 10−4 |
80 |
Portable audio player at maximum level | 10−2 |
100 |
Front rows of rock concert | 10−1 |
110 |
Threshold of pain | 10 | 130 |
Jet aircraft 50 m away | 102 |
140 |
Find the decibel level of a TV that has an intensity of 250×10−7 W/m2
We are given I=250×10−7 W/m2,
So the decibel level of this TV is approximately 74 dB.
Find the decibel level of a TV that has an intensity of 200×10−7W/m2
How much more intense is a 65-dB sound than a 42-dB sound?
So a sound of 65-dB is about 200 times more intense than a sound of 42-dB.
How much more intense is a 75-dB sound than a 55-dB sound?
Calculate the intensity in watts per square meter of a sound of 73 dB.
What is the intensity of a 48-dB sound?
Pitch of a sound is determined by its frequency. For example, on a piano, the pitch of the note A above middle C may be denoted by A440. This means that A vibrates at 440 Hertz (cycles per second). In music, an interval whose higher note frequency is twice that of its lower note is an octave. Two notes that are an octave apart sound essentially “the same,” but one has a higher pitch. For this reason, notes an octave apart are given the same name. An octave jump is usually divided into 12 approximately equal semitones on a log scale. In an equal tempered scale, a semitone is further divided into 100 cents. So there are 1200 cents in an octave.
We note that the pitch difference of 1 octave means f=2f0,
which agrees with our previous assertion.
Most people are not sensitive to pitch differences of less than about 2 cents.
In an “equal tempered” scale for a given reference frequency f0
For the given reference frequency f0,
In an equal tempered scale, the notes are denoted as follows:
If the frequency of A is 440 Hz, estimate the frequency of B.
The next example shows that an equal tempered scale does not always produce the best harmonics.
A jump of one perfect fifth gives a frequency increase of 50%. An equal tempered fifth is 700 cents. Find the difference in cents (rounded to the nearest cent). Is the difference noticeable?
If the reference frequency is f0
Because a fifth on an equal tempered scale is 700 cents, the difference is a little less than 2 cents. This difference is barely noticeable to most listeners.
Repeat Example 10 if a perfect major third gives a frequency increase of 25% and an equal tempered major third is 400 cents.
4 Define magnitude of star brightness.
Two thousand years ago Greek astronomers Hipparchus and Ptolemy created a system to quantify the apparent brightness (wattage received on the surface of the Earth) of stars. They classified star brightness on a scale from magnitude 1 for the brightest visible star to magnitude 6 for the dimmest visible star. Other stars were assigned magnitudes based on their brightness compared with those of magnitude 1 or 6. This system was refined considerably, leading to a much larger range of magnitude values and the following definition.
Although this definition of apparent magnitude was originally intended for comparing the brightness of stars, it can also be used for objects such as planets and their moons. The Hubble telescope can detect stars with apparent magnitude 30.
Compare the brightness of a magnitude 1 star with that of a magnitude 6 star.
Find the magnitude m of a star that is 650 times as bright as one of magnitude 7.25.
m2−m1=2.5 log(b1b2)Apparent magnitude formula6−1=52log(b1b2)m2=6, m1=1, and 2.5=522=log(b1b2)Multiply both sides by 25 and simplify.b1b2=102Exponential formb1=102b2=100b2Multiply both sides by b2.
So a star of magnitude 1 is 100 times brighter than a star of magnitude 6.
Letting m2=m, m1=7.25
Compare the brightness of a magnitude 0 star with that of a magnitude 2 star.
Find the magnitude m of a star that is 50% brighter than a star of magnitude 4.6.
5 Build models from data.
In Section 2.3, we discussed modeling the relationship between two variables (x and y) when the scatterplot of the corresponding data points appears to be linear. The linear least-squares technique (“regression line” y=ax+b
We can apply logarithm transformations to either the y-variable, the x-variable, or both the x- and y-variables at the same time. If we find that one of these transformations results in a linear relationship that produces a regression line, we can deduce that the relationship between the original variables is exponential, logarithmic, or power, respectively. The transformations and resulting models are summarized in the following table:
x variable | y variable | Regression line | Model |
---|---|---|---|
x | y | y=ax+b |
Linear: y=ax+b |
x | ln y | ln y=ax+b |
Exponential: y=ceax, |
ln x | y | y=a ln x +b |
Logarithmic: y=a ln x +b |
ln x | ln y | ln y=a ln x +b |
Power: y=cxa |
The graphical representations of scattered data and the corresponding models are shown in Figure 4.22.
The resulting models can be used to summarize the data, to predict unobserved values, or to understand the mechanisms that produce the observed values.
The data in Table 4.11 show the temperature inside a parked car on a hot day (90° F)
Elapsed Time | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 | 55 | 60 | 65 | 70 | 75 | 80 | 85 | 90 |
Temperature | 101 | 110 | 118 | 122 | 126 | 130 | 133 | 136 | 136 | 138 | 140 | 141 | 142 | 142 | 144 | 144 | 145 | 145 |
ln (Elapsed Time) | 1.61 | 2.30 | 2.71 | 3.00 | 3.22 | 3.40 | 3.56 | 3.69 | 3.81 | 3.91 | 4.01 | 4.09 | 4.17 | 4.25 | 4.32 | 4.38 | 4.44 | 4.50 |
Source: Based on RACQ study.
|
Set x as the elapsed time and y as the temperature inside a car. Apply the logarithmic transformation to x. Create a scatterplot of data (y versus x) as well as a scatterplot of transformed data (y vs. ln x) and find the logarithmic model that fits these data.
The corresponding scatterplots of the data are presented in Figure 4.23.
From the scatterplot of the transformed data, we see that the logarithmic model was a good choice. Using a graphing calculator (see page 208), we find the regression line (in the variables Y=y
From that we find the coefficients of the logarithmic model that fit the original data:
The regression line and logarithmic model are shown on their corresponding scatterplots in Figure 4.23. This type of model is called the logarithmic growth model.
Repeat Example 12 with the data shown in Table 4.12 . Use natural log values for the elapsed time rounded to two decimal places. The data show the temperature inside a parked car in identical conditions as in Example 12 , except now the car windows are left rolled down 2 inches.
Elapsed Time | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 | 55 | 60 | 65 | 70 | 75 | 80 | 85 | 90 |
Temperature | 92 | 98 | 103 | 108 | 111 | 114 | 116 | 118 | 116 | 118 | 121 | 121 | 122 | 122 | 123 | 123 | 123 | 125 |
Source: Based on RACQ study. |
In the next example, we investigate Kleiber’s Law. In 1932, Max Kleiber observed that, for the vast majority of animals, an animal’s metabolic rate q varies directly with the 34
The basal metabolic rate is the amount of energy expended while at rest to support only the functioning of the vital organs.
The data in Table 4.13 show the body weight (in grams, g) and the corresponding basal metabolism (in watts, W) for 15 different species of small animals.
Body Weight (g) | 16 | 67 | 120 | 206 | 336 | 490 | 598 | 828 | 1120 | 1551 | 1782 | 2250 | 2750 | 3257 | 3600 |
Metabolism (W) | 0.13 | 0.33 | 0.58 | 1.08 | 1.19 | 1.24 | 1.78 | 2.21 | 2.84 | 3.23 | 3.01 | 5.07 | 6.81 | 6.05 | 7.74 |
ln (Body Weight) | 2.77 | 4.20 | 4.79 | 5.33 | 5.82 | 6.19 | 6.39 | 6.72 | 7.02 | 7.35 | 7.48 | 7.72 | 7.92 | 8.09 | 8.18 |
ln (Metabolism) | −2.04 |
−1.11 |
−0.545 |
0.077 | 0.174 | 0.215 | 0.577 | 0.793 | 1.04 | 1.17 | 1.10 | 1.62 | 1.92 | 1.80 | 2.05 |
Source: Based on data from “Size and Power ln Mammals” by A. Hausner, J. Exp. Biol. 160, 25–54 (1991). |
Set x as the body weight and y as the metabolism rate. Apply the logarithmic transformation to both x and y. Create a scatterplot of data (y vs. x) as well as a scatterplot of transformed data (ln y vs. ln x) and find the power model that fits these data.
The corresponding scatterplots of the data are shown in Figure 4.24.
From the scatterplot of the transformed data, we see that the power model is a good choice. Using a graphing calculator (see page 208), we find the regression line (in variables Y=ln y
From that, we find the coefficients of the power model that fit the original data:
The regression line and power model are shown with their corresponding scatterplots in Figure 4.24. Note that the power exponent 0.73 we obtain is very close to 34=0.75
Repeat Example 13 with the data shown in Table 4.14 .
Body Weight (kg) | 1.1 | 1.5 | 2.4 | 3.6 | 4.0 | 4.3 | 5.7 | 6.5 | 6.9 | 7.7 | 13.6 | 15.9 | 25 | 30 | 45.2 |
Metabolism (W) | 2.23 | 2.79 | 3.65 | 7.74 | 5.63 | 6.49 | 5.11 | 6.52 | 8.23 | 11.03 | 14.98 | 12.54 | 14.08 | 34.46 | 46.85 |
Source: Based on data from “Size and Power ln Mammals” by Hausner, J. Exp. Biol. 160, 25–54 (1991). |
On the Richter scale the magnitude of an earthquake M=_.
The loudness L of sound of intensity I is given by L=_
If f0
If two stars have magnitudes m1
True or False. If the pH value of a solution is bigger than 7, the solution is acidic.
True or False. The acidity of a solution increases as its pH value increases.
True or False. For a one-unit increase in the magnitude of an earthquake, its intensity increases tenfold.
True or False. The apparent brightness of a star increases if its magnitude decreases.
In Exercises 9–12, the concentration [H+]
[H+]=10−8
[H+]=10−4
[H+]=2.3×10−5
[H+]=4.7×10−9
In Exercises 13–16, use the following information. Negative ions, designated by the notation [OH−]
pH=6
pH=8
pH=9.5
pH=3.7
In Exercises 17–20, the magnitude M of an earthquake is given.
Find the earthquake intensity I in terms of the zero-level earthquake intensity I0.
Find the energy released by the earthquake.
M=5
M=2
M=7.8
M=3.7
In Exercises 21–24, the energy E released by an earthquake is given.
Find the earthquake’s magnitude.
Find the earthquake’s intensity.
E=1013.4 joules
E=1010.4 joules
E=1012 joules
E=109 joules
In Exercises 25–28, the intensity I of a sound is given. Find the loudness L of the sound (I0=10−12 W/m2).
I=10−8 W/m2
I=10−10 W/m2
I=3.5×10−7 W/m2
I=2.37×10−5 W/m2
In Exercises 29–32, a loudness L is given. Find the intensity of the sound.
L=80 dB
L=90 dB
L=64.7 dB
L=37.4 dB
In Exercises 33–36, use the equal tempered scale with the twelve semitones A, A#, B, C, C#, D, D#, E, F, F#, G, and G# and the frequency of A is 440 Hz. On a piano keyboard, the distance between two white keys that are side-by-side is a whole note if there is a black key between them and is a semitone if there is no black key between them. A black key (such as the one between C and D) is called a sharp. The black key between C and D is denoted by C#.
Remember that the change in frequency P(f) between any two consecutive semitones is 100 cents.
Find the frequencies of A# and C.
Find the frequencies of D and E.
A whole tone is a frequency ratio of 10:9. That is, ff0=109.
Repeat Exercise 35 if a whole tone is a frequency ratio of 9 : 8.
The magnitudes of two stars, A and B, are 4 and 20, respectively. Compare the brightness of these stars.
Repeat Exercise 37 for two stars with magnitudes −2
The magnitude of star A is 2 more than that of star B. How is their corresponding brightness related?
The magnitude of star A is 1 more than that of star B. How is their corresponding brightness related?
pH of human blood. The hydrogen ion concentration [H+]
pH of milk. The hydrogen ion concentration [H+]
pH of some common substances. The pH value of a sample of each substance is given. Find [H+]
|
pH=3.15 |
|
pH=7.2 |
|
pH=7.78 |
|
pH=3 |
pH of some common substances. The pH value of a sample of each substance is given. Find [H+]
|
pH=1.0 |
|
pH=8.7 |
|
pH=10.6 |
|
pH=2.3 |
Acid rain. In the Netherlands, the average pH value of the rainfall is 3.8. Find the average concentration of hydrogen ions [H+]
pH of a solution. Suppose the pH value of a solution A is 1.0 more than the pH value of a solution B. How are the concentrations of hydrogen ions in the two solutions related?
Comparing pH of solutions. Suppose the hydrogen ion concentration in a solution A is 100 times that in solution B. How are the pH values of the two solutions related?
pH of a solution. Suppose the pH value of a solution is increased by 1.5. How much change does this represent in the hydrogen ion concentration of this solution? Does the increase in pH make the solution more acidic or more basic?
pH of a solution. Suppose the hydrogen ion concentration of a solution is increased 50 times. How much change does this represent in the pH value of this solution? Does this increase in [H+]
China earthquake. The Great China Earthquake of 1920 registered 8.6 on the Richter scale.
What was the intensity of this earthquake?
How many joules of energy were released?
San Francisco earthquake. Repeat Exercise 50 for the Great San Francisco Earthquake of 1906, which registered 7.8 on the Richter scale.
Comparing earthquakes. Table 4.9 lists two earthquakes in Japan.
Compare the intensities of these earthquakes.
Compare the energies released by these quakes.
Comparing earthquakes. Suppose earthquake A registers 1 more point on the Richter scale than earthquake B.
How are their corresponding intensities related?
How are their released energies related?
Comparing earthquakes. Repeat Exercise 53, if earthquake A registers 1.5 more points on the Richter scale than earthquake B.
Comparing magnitudes. If one earthquake is 150 times as intense as another, what is the difference in the Richter scale readings of the two earthquakes?
Comparing magnitudes. If the energy released by one earthquake is 150 times that of another, what is the difference in the Richter scale readings of the two earthquakes?
In Exercises 57–64, use Table 4.10 as necessary.
dB and intensity. What is the loudness, in decibels, of a radio with intensity 5.2 ×10−5 W/m2
dB of a jet. What is the loudness of a jet aircraft with intensity 2.5×102 W/m2
Comparing two sounds. How much more intense is a sound of the threshold of pain (130 dB) than a conversation at 65 dB?
Comparing two sounds. How much more intense is a 75 dB sound than a 62 dB sound?
Comparing two sounds. Suppose a sound is 1000 times as intense as one at the threshold of pain for the human ear. Find the loudness of this sound in decibels.
Comparing intensities. Show that if two sounds of intensity I1
Comparing intensities. Suppose one sound registers 1 decibel more than another. How are the intensities of the two sounds related?
Comparing sounds. The noise level of a street sound outside a new office building in downtown Chicago is measured to be approximately 70 dB. With special insulation material, the noise level inside the office building is reduced to 29 dB. Compare the intensities of sounds outside and inside the building.
Violin tuning. A violinist is tuning her A-string to match concert pitch. The concert’s A is a frequency of 440 Hz. Her A-string is improperly tuned to 441 Hz. Find the difference in cents.
Violin tuning. Repeat Exercise 65 if her A-string is improperly tuned to 438 Hz.
In Exercises 67–72, use the following table of approximate apparent magnitudes of some celestial objects.
Object | The Sun | Full Moon | Venus | Sirius | Vega | Saturn | Neptune |
---|---|---|---|---|---|---|---|
Magnitude | −27 |
−13 |
−4 |
−1 |
0.03 | 1.47 | 7.8 |
Sun-moon How many times brighter is the sun than the full moon?
Moon-Vega. How many times brighter is the full moon than Vega?
Sun-Venus. How many times brighter is the sun than Venus?
Venus-Neptune. How many times brighter is Venus than Neptune?
Saturn-Star. What is the magnitude of a star that is 560 times brighter than Saturn?
Neptune-Star. What is the magnitude of a star that is 1 billion times brighter than Neptune?
Solar panels. The data in the following table show the number of annual solar PV cells installations (in megawatts [MW]) over the period 2004–2014.
Year | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 |
PV Installations | 58 | 79 | 105 | 160 | 298 | 382 | 852 | 1922 | 3369 | 4776 | 6201 |
Source: Solar Energy Industries Association, www.seia.org |
Find the exponential model that fits the data, letting the year 2004 correspond to x=1
CD sales. The data in this table show annual CD album sales in the United States (in millions of units) over the period, 2004–2014.
Year | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 |
CD Sales | 651 | 599 | 553 | 449 | 361 | 295 | 237 | 224 | 193 | 165 | 141 |
Source: Nielsen Soundscan. |
Find the exponential model that fits the data, letting the year 2004 correspond to x=1
NFL draft. The data in the following table show the estimated Career Value Rating in relation to the Draft Pick Number for the NFL averaged over the past 25 years.
Pick Number | 1 | 5 | 10 | 15 | 20 | 25 | 30 | 40 | 50 | 100 | 200 |
Career Value Rating | 64.8 | 51.4 | 44 | 38.2 | 36 | 32.1 | 33.1 | 27.7 | 27.4 | 18.2 | 10.5 |
Find the logarithmic model that fits the data. This type of model is called the logarithmic decay model.
Heart rate. The data in this table show the body weight (in grams g) and corresponding heart rate (in beats per minute) for nine different species of warm-blooded animals at rest.
Weight | 25 | 60 | 200 | 341 | 1100 | 2000 | 5000 | 90000 |
Heart Rate | 670 | 450 | 420 | 378 | 190 | 150 | 120 | 60 |
Find the power model that fits the data. This type of model is called the power decay model.
Decibel levels. The decibel level of Professor Stout’s voice decreases with the distance from the professor according to the formula
where L is the decibel level and r is the distance in feet from the professor to the listener.
Find the decibel levels at distances of 10, 25, 50, and 100 feet.
How far must a listener be from the professor so that the decibel level drops to 0?
Express L in the form L=a+b log r,
Express r as a function of L.
Sound intensity. The intensity of sound is inversely proportional to the square of the distance. That is, I=Kr2,
Show that if L1
A military jet at 30 meters has 140 decibel level of loudness. How loud will you hear this jet sound at a distance of 100 meters? 300 meters?
Express r2r1
Nuclear bomb. Compare the energy released by a 1 megaton nuclear bomb (about 5×1015
Adrian was calculating the pitch of a sound and mistakenly transposed the frequency f of the sound with the reference frequency f0.
If the A-string of a violin is tuned to 880 Hz with an error of ±10
Show that the formula for star brightness can be written as b1b2=100.4(m2−m1).
A star of magnitude m is 176 times brighter than a star of magnitude 3.42. Find m.
Star luminosity. The luminosity L of a star is the power output at a star’s surface (measured in watts). The star’s apparent brightness b is the wattage received on Earth’s surface. The apparent magnitude m of a star depends on its absolute magnitude M (related to L) and its distance D in parsecs (1 parsec≈3.26 light years)
Given m and M for a star, show that its distance D from Earth is D=101+(m−M)/5
For the star Alpha Centauri, m=0
Compare the luminosity and brightness of Alpha Centauri.
In Exercises 85–91, find the value of a variable that satisfies the given condition.
Find x if 3x+2y=7
Find x if −2x+9y=5
Find y if −4x+7y=7
Find y if 23x−14y=−5
Find the slope of the line with equation 10x−2y=28.
Find the slope of the line with equation 15x+5y=2.
Find an equation of the line through the point (5, −1)
Find an equation of the line through the point (3, 3) parallel to the line with equation x+2y=1.
Find values for a and b so that the equation ax+by=3