A log scale relating x and y is shown in the table of values below:
| \text{Log scale measure } (y) | \text{Linear measure } (x) | |
|---|---|---|
| 0 | = | 1 |
| 1 | = | 10 |
| 2 | = | 100 |
| 3 | = | 1000 |
| 4 | = | 10\,000 |
Form an equation relating x and y. Express the equation in logarithmic form.
The number line below has a \log_{10} scale. Plot the number 100 on the scale below:
Each number line below has a \log_{10} scale. For each number line, determine the value of the point plotted on the line. Round your answers to two significant figures.
The given histogram shows the area (in square kilometres) for 12 countries using a \log_{10} scale on the x-axis:
How many countries have an area of between 10^5 and 10^6 square kilometres?
How many countries have an area of between 10\,000\text{ km}^2 and 100\,000\text{ km}^2?
The histogram has been plotted on a \log_{10} scale:
How many scores are between 10^0 and 10^1?
What is the total frequency of scores between 1 and 100?
What is the total frequency of scores larger than 0.1?
What is the total frequency of scores smaller than 10?
The histogram shows the masses (in grams) of a group of insects and animals, plotted on a \log_{10} scale:
How many animals and insects have a mass between 10^1 and 10^2 grams?
How many animals and insects are included in the data altogether?
A caterpillar has a mass of 3 grams. What value does this take on a \log_{10} scale? Round your answer to two decimal places.
Which interval on the histogram would include a cat that has a mass of 4.5\text{ kg}?
From the data in the histogram, how many of the animals or insects had a mass that was over 100 grams?
The histogram below shows the weights (in kilograms) of 26 zoo animals plotted on a log scale:
A monkey has a weight of 45.8\text{ kg}. What is the \log_{10} of 45.8 correct to two significant figures?
What weight (in \text{kg}) does the number - 2 represent on the log weights scale?
How many animals have a weight of at least 1000\text{ kg}?
What percentage of animals have a weight less than 0.1 \text{ kg}? Round your answer to two significant figures.
How many animals have a weight of at least 0.1\text{ kg} but less than 100\text{ kg}?
The histogram below shows the income (in dollars) of 24 employees plotted on a log scale:
A employee has an income of \$46\,974. What is the \log_{10} of 46\,974 correct to two significant figures?
What income does the number 5 represent on the log-income scale?
How many people have an income of at least \$10\,000?
What percentage of people have an income less than \$100\,000?
How many people have an income of at least \$10\,000 but less than \$10\,000\,000?
A financial planner has clients with a large range of annual incomes. To manage the data more easily, the income (in dollars) of each client is first converted to a \log_{10} scale.
What income would result in 4.9 on a \log_{10} scale?
What range of incomes corresponds to a range of 5 to 6 on the \log_{10} scale?
The lifespans of various animals and insects are shown in the table below:
| \text{Animal/Insect} | \text{Lifespan (days)} | \text{Lifespan in a log}_{10} \text{ scale} |
|---|---|---|
| \text{Ant} | 20 | |
| \text{African elephant} | 25\,500 | 4.41 |
| \text{Bowhead whale} | 71\,400 | |
| \text{Bumblebee} | 21 | 1.32 |
| \text{Chicken} | 2920 | 3.47 |
| \text{Dog} | 5110 | |
| \text{Emperor penguin} | 8500 | 3.93 |
| \text{Fly} | 18 | |
| \text{Giraffe} | 10\,120 | |
| \text{Hamster} | 780 | 2.89 |
| \text{Ladybug} | 365 | |
| \text{Little penguin} | 2190 | 3.34 |
| \text{Monkey} | 7300 | |
| \text{Shark} | 10\,980 | 4.04 |
Convert the lifespans to a \log_{10} scale to complete the table, rounding each value to two decimal places. Some values have already been filled in.
Create a histogram to represent this data, on a \log_{10} scale.
The table below lists the oil production in barrels per day for 20 countries:
| \text{Country} | \text{Oil Production}\\ \text{(barrels per day)} | \text{log}_{10}\text{ Oil Production} |
|---|---|---|
| \text{Saudi Arabia} | 10\,625\,000 | |
| \text{China} | 3\,938\,000 | |
| \text{Norway} | 1\,763\,000 | |
| \text{India} | 736\,000 | |
| \text{Ecuador} | 555\,000 | |
| \text{Thailand} | 265\,000 | |
| \text{Brunei} | 113\,000 | |
| \text{Uzbekistan} | 85\,000 | |
| \text{France} | 61\,000 | |
| \text{Peru} | 39\,000 | |
| \text{Albania} | 21\,000 | |
| \text{Croatia} | 18\,000 | |
| \text{Belgium} | 13\,000 | |
| \text{Finland} | 13\,700 | |
| \text{Portugal} | 7100 | |
| \text{Bangladesh} | 4800 | |
| \text{Japan} | 4000 | |
| \text{Lativia} | 1150 | |
| \text{Costa Rica} | 300 | |
| \text{Zambia} | 200 |
Calculate the \log_{10} value of each country's oil production to complete the table.
Construct a frequency histogram of log oil production.
The number of registered nurses working in hospitals t years after the year 2002 can be modelled by the equation N = 28 \log_{4} \left(t + 2\right), where N represents the number of nurses in thousands.
How many registered nurses were working in hospitals in the year 2002?
Find the number of registered nurses in 2004.
A major communications company found that the more they spend on advertising, the higher their revenue. Their sales revenue R, where x represents the amount they spend on advertising ( R and x in thousands of dollars) is given by:
R = 10 + 20 \log_{4} \left(x + 1\right)Determine their sales revenue if they spend no money on advertising.
Determine their sales revenue if they spend \$14\,000 on advertising. Round your answer to the nearest thousand.
Would you say that every extra \$1000 spent on advertising becomes more or less effective in terms of raising revenue?
As elevation A (in metres) increases, atmospheric air pressure P (in pascals) decreases according to the equation:
A = 15\,200 \left(5 - \log P\right)Trekkers are attempting to reach the 8850\text{ m} elevation of Mt. Everest’s summit. When they set up camp at night, their barometer shows a reading of 45\,611 \text{ Pa}. How many more vertical metres do they need to ascend to reach the summit? Round your answer to the nearest metre.
The Richter Scale is a base 10 logarithmic scale used to measure the magnitude of an earthquake, given by R = \log_{10} s, where s is the relative strength of the quake.
The aftershock of an earthquake measured 6.7 on the Richter Scale, and the main quake was 4 times stronger. Find the magnitude of the main quake on the Richter Scale, to one decimal place.
The decibel scale, which is used to record the loudness of sound, is a logarithmic scale.
In the decibel scale, the lowest audible sound is assigned the value of 0.
A sound that is 10 times louder than 0 is assigned a decibel value of 10.
A sound 100 (10^{2}) times louder than 0 is assigned a decibel value of 20.
In general, an increase of 10 decibels corresponds to an increase in magnitude of 10. The table shows the decibel value for various types of noise:
How many times louder than 0 is the sound of a bedroom?
How many times louder than 0 is the sound of an office? Write your answer as a power of 10.
If the sound of a normal speaking voice is 50 decibels, and the sound in a bus terminal is 80 decibels, how many times louder is the bus terminal compared to the speaking voice?
How many times louder is the sound of industrial noise than the sound of a wind turbine?
| Type of noise | Decibel value |
|---|---|
| \text{Jet plane} | 150 |
| \text{Pneumatic drill} | 120 |
| \text{Industrial} | 110 |
| \text{Stereo music} | 100 |
| \text{Inside a car} | 90 |
| \text{Office} | 70 |
| \text{Household} | 60 |
| \text{Wind turbine} | 50 |
| \text{Bedroom} | 30 |
| \text{Falling leaves} | 20 |
The sound level or loudness, L, of a noise is measured in decibels \text{(dB)}, and is given by the formula: L = 10 \log \left(\dfrac{I}{A}\right), where I (in \text{watts/cm}^{2}) is the intensity of a particular noise and A is the the intensity of background noise that can barely be heard.
At a concert, standing near a speaker exposes you to noise that has intensity of about \\I = 0.5 \times 10^{13} A.
How many decibels is this? Round your answer to the nearest \text{dB}.
Noises measuring up to 85\text{ dB} are harmless without ear protection. By how many decibels does the noise at a concert exceed this safe limit? Round your answer to the nearest \text{dB}.
The maximum intensity which the human ear can handle is about 120 \text{ dB}. The noise in a recycling factory reaches 132.9 \text{ dB}. How many times louder than the maximum intensity is the factory noise? Round your answer to one decimal place.
If one person talks at a sound level of 60 \text{ dB}, find the value of L which represents the decibel level of 100 people, each talking at the same intensity as that one person.
If a sound intensity doubles, by how much does the level of sound in decibels increase?
Given A=10^{-16} \text{ watts/cm}^{2}, find:
The sound level of a sound with intensity I = 10^{ - 5 }\text{ watts/cm}^{2}.
The sound intensity of a passenger plane passing over houses prior to landing, if the engine’s loudness is registered at 103 \text{ dB}. Give an exact answer.
Search engines give every web page on the internet a score (called a Page Rank) which is a rough measure of popularity/importance. One such search engine uses a logarithmic scale so that the Page Rank is given by: R = \log_{11} x, where x is the number of views in the last week.
Determine the Page Rank of a web page that received 7300 views in the last week. Round your answer to the nearest integer.
Google uses a base-10 logarithmic scale to get a web page’s Page Rank: R = \log_{10} x. How many more times the views did a web page with a Page Rank of 5 get, than one with a Page Rank of 3?
The time taken (t years) for A grams of a radioactive substance to decompose down to y grams, where k is a constant related to the properties of a particular substance is given by:
t = - \dfrac{1}{k} \log_{2.3} \left(\dfrac{y}{A}\right)Consider the following substances, giving your answers to the nearest year:
R_{1} has a constant of k=0.000\,43
R_{2} has a constant of k=0.000\,47
Find how long it takes 140 units of R_{1} to decompose down to 105 units.
Find how long it takes 200 units of R_{2} to decompose down to 150 units.
Find the half-life of each substance. That is, determine how long it takes a quantity of a substance to decompose down to half the original quantity.
Will it take twice the half-life for each substance to decompose completely? Explain your answer.
Consider the function y = 8 \left(2\right)^{ 2 x}, for x \geq 0.
The function above can be written as \log_{2} y = m x + k. Find the values of m and k.
Sketch the graph of \log_{2} y against x.
Find the rate of change of the linear function.
Find the value of x when y = 16.
The Palermo impact hazard scale is used to rate the potential for collision of an object near Earth. The hazard rating P is given by the equation P = \log R, where R represents the relative risk of collision. Two asteroids are identified as having a relative risk of collision of \dfrac{6}{7} and \dfrac{4}{5} respectively. Find the exact difference in their measure on the Palermo impact scale.
pH is a measure of how acidic or alkaline a substance is. The pH \left(p\right) of a substance can be found according to the formula: p = - \log_{10} h, where h is the substance’s hydrogen ion concentration.
The pH scale goes from 0 to 14, with 0 being most acidic, 14 being most alkaline and pure water has a neutral pH of 7.
Store-bought apple juice has a hydrogen ion concentration of about h = 0.0002. Find the pH of the apple juice correct to one decimal place.
Is the apple juice acidic or alkaline?
A banana has a pH of about 8.3. Find h, its hydrogen ion concentration. Give your answer as an exact value.
Researchers conducted a test to determine how well information is retained through the method of rote learning. To do this, they asked students to memorise mathematical formulae in the lead up to the first test, and then study no further. They continued to test them once a month over 7 months. They found that the average student’s test scores (P) decreased over time (t months), but at a slowing rate.
Which equation could be used to model their findings of the relationship between P and t under the rote learning method?
P = - 86 - 23 \log_{2} \left(t + 1\right)
P = 23 \log_{2} \left(t + 1\right) - 86
P = 86 - 23 \log_{2} \left(t + 1\right)
P = 23 \log_{2} \left(t + 1\right) + 86
Using the equation in part (a) to model student performance over time, find the average student’s test score on the initial test at t = 0.
Find the average student’s test score on the 6th test. Round your answer to the nearest whole number.
Find the number of months it takes for the average student’s test score to equal 0 using the rote learning method. Write your answer as a whole number of months.
The information entropy H (in bits) of a randomly generated password consisting of L characters is given by: H = L \log_{2} N where N is the number of possible symbols for a character in the password.
A case sensitive password consisting of seven characters is to be made using letters from the alphabet and/or numerical digits.
Find the value of L.
Find the value of N.
Find the value of H correct to two decimal places.
It was found that a seven character password resulted in an entropy of 28 bits. Find the possible number of symbols for a character.
A cent, n, is a logarithmic unit of measurement for musical intervals, and measures the ratio between two frequencies a and b of two different notes such that a \geq b. There are 1200 cents in an octave and the ratio is given by:
\dfrac{a}{b} = 2^{\frac{n}{1200}}Rearrange the equation to make n the subject.
Calculate the number of cents between notes that have frequencies of 800 and 600.
If a note has a frequency of 500 and the next note is 700 cents larger, find the frequency of the second note. Round your answer to the nearest whole number.
When we know the apparent magnitude of an object m, and the distance d to the object measured in parsecs, we can calculate the absolute magnitude, M, according to:
m - M = 5 \left(\log d - 1\right)
Calculate the distance in parsecs of Betelgeuse from Earth if Betelgeuse has an apparent magnitude of 0.5 and an absolute magnitude of - 5.8. Round your answers to one decimal place.
The Supernova of the year 1006 is considered the brightest stellar event ever observed. The apparent magnitude of this event was - 7.5. If the distance of the event from Earth was 7200 light years away, and 10 parsecs is the equivalent of 32.6 light years, find the absolute magnitude of Supernova 1006. Round your answer to one decimal place.
When comparing the ratio in the brightness between two objects in the night sky, we can use their apparent magnitudes, m_1 and m_{\text{ref}}, and respective brightness intensities I_1 and I_{\text{ref}} such that:
m_1 - m_{\text{ref}}=-2.5 \log_{10} \dfrac{I_1}{I_{\text{ref}}}Betelgeuse has an apparent magnitude of 0.5 while Alpha Centauri has an apparent magnitude of 0.0. Calculate how many times brighter Alpha Centauri appears compared with Betelgeuse, correct to two decimal places.
The ratio in brightness between the Sun and a full Moon is approximately 400\,000. If the Sun has an apparent magnitude of -26.7, what is the apparent magnitude of the full Moon? Round your answer to two decimal places.
Consider the formula d = 10 \log \left(\dfrac{P}{k}\right), for some constant k.
If P is increased by a factor of 10, by how much does d increase?
If P is increased by a factor of 5, by how much does d increase? Write your answer as an exact value.
Determine the value of P for which d = 0.