6.06 Applications of logarithms and exponentials
Exponential and logarithmic functions are used to model a wide variety of behaviours in the real world. Applications of these functions can be in very different disciplines but the mathematics of each remains essentially the same. The examples that follow are just some of the many applications in science, technology and economics.
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.
Applications of exponential functions: growth and decay
Two quantities have an exponential relationship if, whenever the input changes by a uniform interval, the output is multiplied by a constant factor. Consequently, this simple relationship describes many situations such as the value of a term deposit in a savings account or the growth of a population. We refer to these situations as examples of exponential growth and decay.
Here are some examples:
Finance: The value of a term deposit $P$P in dollars at time $t$t given in years can be given by the equation:
$P=A\left(1+r\right)^t$P=A(1+r)t
where $A$A is the initial term deposit and $r$r is the annual interest rate. As we can see, $P$P grows exponentially since for every year that passes, we multiply the previous term deposit by the constant factor of $1+r$1+r
Bacteria growth: Exponential relationships also arise in biology. We might wish to count the total number of bacteria cells that have been cultivated on a petri dish. If there is initially $1$1 live cell and $2$2 dead cells, how can we find the total number of cells after $t$t days given that the live cells double every day? The total number of cells $T$T after $t$t days can be given by the equation:
$T=2^t+2$T=2t+2
where the number of live cells is $2^t$2t and the number of dead cells remains constant at a value of $2$2.
Population change: Quickly growing populations can be modelled with exponentials. For example, we might begin with only ten animals of a particular species but observe that the number of animals doubles every month. Thus, if there are $n$n individuals at a given time, there will be $2n$2n in another month. If $t$t denotes the amount of time that has passed in months, then we can construct an equation relating $n$n and $t$t as follows:
$n=10\left(2^t\right)$n=10(2t)
For every month that passes, we multiply the previous value of $n$n by two. Of course when $t=0$t=0, we get the initial number of animals, that being $n=10$n=10.
Practice questions
Question 4
The number of fungal cells, $N$N, in a colony after $t$t hours is given by the equation $N=5000\left(4^t\right)$N=5000(4t).
Determine the initial population of fungal cells.
Determine the population of fungal cells after $8$8 hours.
Plot the graph of fungal cell population over time.
According to the graph, approximately how many hours will it take for the population to reach $3$3 times the original population?
Question 5
Peter received a lump sum payment of $\$50000$$50000 for an insurance claim (and decided not to put it in a savings account).
Every month, he withdraws $5%$5% of the remaining funds.
The funds after $x$x months is shown below.
How much would the first withdrawal be?
How much would be left after the first withdrawal?
According to the graph, which of the following is the best estimate for the amount left after one year?
$\$0$$0
A$\$3000$$3000
B$\$27000$$27000
C$\$30000$$30000
DAccording to the graph, which of the following is the best estimate for the number of months until the amount reaches $\$30000$$30000?
$5$5
A$10$10
B$20$20
C$40$40
D
Question 6
A new appliance is valued at $\$990$$990.
Each year it is worth $9%$9% less than the previous year's value.
Calculate the value of the appliance after the first year to the nearest cent.
Calculate the value of the appliance after the second year to the nearest cent.
Determine the equation that relates the value of the appliance, $A$A, with the number of years passed, $t$t.
Using the equation in part (c), calculate the value of the appliance after twelve years to the nearest cent.