Applications of logarithmic functions
There are many things that can be measured on a logarithmic scale. Rather than each step up on a regular linear scale involving a change in terms of a common difference (i.e. from $2$2 to $3$3 is the same change as from $4$4 to $5$5), in log scales each step up involves a change based on a common ratio.
Below we can see numbers on a log scale measured in factors of $10$10. That is, a measure of $5$5 on a log scale is $10$10 times greater than $4$4. The diagram below illustrates a log scale compared to a linear scale:
Richter scale: The magnitude of an earthquake (how much energy it releases) can be measured on a logarithmic scale called the Richter scale. This is calculated using information gathered from measuring devices called seismographs. The Richter scale is a base $10$10 logarithmic scale so, for example, an earthquake that measures $4.0$4.0 on the Richter scale is $10$10 times larger than one that measures $3.0$3.0. The scale ranges from $2$2 to $10$10. An earthquake registering below $5$5 is considered minor and anything that registers above $5$5 is considered more severe.
Sound intensity: Sound is measured using a unit called a decibel (dB). Decibels are measured on a log scale where the logarithm involved compares the power level of a sound to the power level of the softest sound a human ear can hear.
The formula can be expressed as: $I=10\log_{10}\left[\frac{P}{P_0}\right]$I=10log10[PP0], where $I$I is the intensity in terms of decibels.
The pH scale: The pH of a solution measures its acidity. The term "pH" originates from Latin and is an acronym for "potentia hydrogenii" - the power of hydrogen. The pH scale is commonly used to represent hydrogen ion activity. It is also a base $10$10 log scale ranging from $0$0 (acid) to $14$14 (base or alkaline). Here, the hydrogen ion activity of pH $4$4 is $10$10 times greater than pH $5$5. A pH of $7$7 is considered neutral (neither acid nor base). Pure water has a pH of $7$7.
Practice questions
Question 1
The decibel scale, used to record the loudness of sound, is a logarithmic scale. The lowest audible sound, with intensity $10^{-12}$10−12 watts/m2 is assigned the value of $0$0. A sound that is $10$10 times louder than this is assigned a decibel value of $10$10. A sound $100$100 ($10^2$102) times louder is assigned a decibel value of $20$20, and so on. In general, an increase of $10$10 decibels corresponds to an increase in magnitude of $10$10.
If the sound of a normal speaking voice is $50$50 decibels, and the sound in a bus terminal is $80$80 decibels, then how many times louder is the bus terminal compared to the speaking voice?
Give your final answer as a basic numeral, not in exponential form.
Question 2
pH is a measure of how acidic or alkaline a substance is, and the pH scale goes from $0$0 to $14$14, $0$0 being most acidic and $14$14 being most alkaline. Water in a stream has a neutral pH of about $7$7. The pH $\left(p\right)$(p) of a substance can be found according to the formula $p=-\log_{10}h$p=−log10h, where $h$h is the substance’s hydrogen ion concentration.
Store-bought apple juice has a hydrogen ion concentration of about $h=0.0002$h=0.0002.
Determine the pH of the apple juice correct to one decimal place.
Is the apple juice acidic or alkaline?
Acidic
AAlkaline
BA banana has a pH of about $8.3$8.3.
Solve for $h$h, its hydrogen ion concentration, leaving your answer as an exact value.
Question 3
The Richter Scale is a base-$10$10 logarithmic scale used to measure the magnitude of an earthquake, given by $R=\log_{10}x$R=log10x, where $x$x is the relative strength of the quake. This means an earthquake that measures $4.0$4.0 on the Richter Scale will be $10$10 times stronger than one that measures $3.0$3.0.
The aftershock of an earthquake measured $6.7$6.7 on the Richter Scale, and the main quake was $4$4 times stronger. Solve for $r$r, the magnitude of the main quake on the Richter Scale, to one decimal place.