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 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.
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{ pascals}. 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 used to measure the intensity of earthquakes. The formula for the Richter scale rating of a quake is given by: R = \log x - \log a, where a is the intensity of a minimal quake that can barely be detected, and x is a multiple of the minimal quake’s intensity.
The given table shows how quakes are categorised according to their Richter scale rating:
Rewrite the formula as a single lograithm.
A seismograph measures the intensity of a quake to be x = 5711 a.
Find the Richter scale rating R of this quake to one decimal place.
In which category does the quake fall?
| Richter rating | |
|---|---|
| Minor | 2-3.9 |
| Light | 4-4.9 |
| Moderate | 5-5.9 |
| Strong | 6-6.9 |
A seismograph measures the intensity of an earthquake to be 15\, 850 times the intensity of a minimal quake.
Find the Richter scale rating R of this quake to one decimal place.
In which category does the quake fall?
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. Some past earthquakes and their Richter scales are shown in the table below:
How many times stronger is a quake of 7.6 than one of 5.2? Round your answer to the nearest whole number.
| Earthquake | Richter scale |
|---|---|
| \text{Sumatra 2004} | 9.1 |
| \text{China 2008} | 7.9 |
| \text{Haiti 2010} | 7.0 |
| \text{Italy 2009} | 6.3 |
How many times stronger was the earthquake in Sumatra compared to the earthquake in China? Round your answer to the nearest whole number.
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 is the sound of industrial noise than the sound of a wind turbine?
If the sound of a normal speaking voice is 50 decibels, and the sound in a bus terminal is 80 decibels, then how many times louder is the bus terminal compared to the speaking voice?
| 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{Houehold} | 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 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 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.
The time taken (t years) for A grams of a radioactive substance to decompose down to y grams is given by: t = - \dfrac{1}{k} \log_{2.3} \left(\dfrac{y}{A}\right), where k is a constant related to the properties of a particular substance.
Consider the following substances, giving your answers to the nearest year:
R_{1} has a constant of k=0.00043
R_{2} has a constant of k=0.00047
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.
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.
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.
The signal ratio D (in decibels) of an electronic system is given by the formula: \\ D = 10 \log \left(\dfrac{F}{I}\right), where F and I are the output and input powers of the system respectively.
Find the input power I in megawatts if the output power is equal to 10\text{ MW} and the signal ratio is 20 decibels.
The given graph shows the equation \\ D = 10 \log \left(\dfrac{F}{5}\right), where the input power is 5\text{ MW}. State the interval that contains:
The signal ratio when F = 9.
The output power when the signal ratio is D = 1.
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. Solve for the values of m and k.
Graph \log_{2} y against x.
Find the rate of change of the linear function.
Solve for x when y = 16.