Final Remarks on Exponential and Logarithmic Models

In this page we will mostly show long examples and questions that require us to combine some of our knowledge of exponential and logarithmic functions, including using a calculator to solve equations and find specific values.

The main skill required to solve the examples/exercises below is understanding how to translate instructions into mathematical models.

The examples/exercises below can be summarised with the following skills:

Finding either an $x$ or $y$ value when being given the other.
Putting some value $f(x)$ into the context of a problem.
Converting between models or understanding a rate of growth.
Understanding and calculating percentage change.
Translating the concept of “rate of change” into a derivative value.
Using properties of powers.
Converting between units.

Many of these can be simplified by knowing how to use your calculator correctly. Using a graphic calculator, we can understand $x$ -values when given $y$ -values. Using the calculator, we can easily replace an $x$ -value into a function $f(x)$ to obtain its corresponding $y$ -value. We can even immediately find the derivative of a function at a specific point, i.e.: the rate of change given some $x$ .

Longer questions will require us to combine these ideas and use the calculator heavily. Shorter questions will typically allow no calculator but rely on your ability to do basic arithmetic and use properties well.

Table of Contents

Examples With Calculator

Example: A laboratory is monitoring a block of dry ice sublimation rate under standard ambient conditions. Its height is given by a function $h$ , defined by:

$h(t) = 60e^{-0.05t + 0.30}$

where $h(t)$ is in cm and $t$ is the number of hours after 08:00.

Calculate the time at which the height of the block goes below $40\text{ cm}$ . Give your answer to the nearest minute.
Calculate the percentage of the height at 08:00 which remains at 16:00.
Determine the rate at which the height of the block is decreasing at 08:00, giving your answer with appropriate units.
The main tracking lab session begins at 10:00. Consider a function $g$ defined by $g(x) = C \cdot A^x$ , where $g(x)$ corresponds to the height of the block in cm $x$ hours after 10:00. Determine the value of $C$ to 1 decimal place and the value of $A$ to 3 decimal places.

As you can see, most of the exercise is understanding mathematical language, applying short mathematical operations (replacing a value, manipulating a term, finding a derivative), translating the instruction into a mathematical operation, interpreting the result in some form. Depending on the calculator available to you, Step 3 could be done instantly by just asking it to find $h'(0)$ and writing out the result. For completeness, the more and clearer you write things, the better.

Example: During a severe storm, a lightning strike knocks out the primary solar array of an automated alpine weather station. The station must run entirely on its internal emergency battery banks. A repair technician is scheduled to arrive via helicopter exactly $3\text{ days}$ after the strike.

The energy consumed by the station’s core climate sensors each day is modeled by a function $C$ , defined by:

$C(n) = 3.15e^{0.024n}$

where $n$ is the number of complete days after the solar grid failure and $C(n)$ is the electrical energy consumed on the $n\text{-th}$ day in kilowatt-hours (kWh).

Determine the percentage change from one day to the next in the weather station’s energy consumption.
Calculate the station’s energy consumption in kilowatt-hours on the first day.

The remaining stored energy in the battery bank in kilowatt-hours after a strike is given by the function:

$R(t) = 180 - 145e^{0.024t}$

where $t$ is the elapsed time measured in days since the initial lightning strike.

Calculate the stored energy reserve in kilowatt-hours immediately after the lightning strike.
Determine whether the energy reserve was a cause for concern during this technician dispatch window.

Examples Without Calculator

Example: A forestry department monitors two different insect populations affecting an oak tree plantation during the summer season.

The number of caterpillars infesting a particular oak tree is modeled by the function $N$ , defined by:
$N(t) = 8 \cdot e^{\ln(1.5) \cdot t}$
where $t$ counts the number of days, with $t = 0$ representing the start of the monitoring period. Find the exact number of caterpillars present at the start of the study ( $t = 0$ ) and exactly $1\text{ day}$ later ( $t = 1$ ).
The number of spider mites on the same tree is modeled by the function $M$ , defined by:
$M(t) = 450 \cdot 0.82^t$
where $t$ counts the number of days as defined above. Determine whether the population of spider mites is expanding or contracting, and state the exact daily percentage change in their numbers.

As you can see, the calculations and even the exponential model used are very different when a calculator is not allowed. It should typically be a matter of simple arithmetic.

Example: Environmental scientists monitor the water reserves of an alpine reservoir during a severe seasonal drought. A study tracking data shows that the total volume of water stored in the reservoir decreases by $20\%$ each month.

The initial water volume at the start of the drought is $50\text{ million m}^3$ .

Justify why the remaining water volume can be modeled by a function $V$ defined by $V(t) = 50 \cdot 0.8^t$ , where $t$ is the number of months since the start of the drought and $V(t)$ is expressed in $\text{million m}^3$ .
Using this model, calculate the exact predicted water volume remaining in the reservoir after $2\text{ months}$ .
Assuming the environmental conditions remain unchanged, evaluate the long-term validity of this model. What happens to the reservoir volume as $t$ approaches a very large number of months, and does this model remain realistic?

Exercises

A pharmaceutical lab measures the thermal degradation of a liquid enzyme solution stored at room temperature. Its total concentration is tracked by a function $C$ , defined by:

$C(t) = 80e^{-0.04t + 0.45}$

where $C(t)$ is the concentration in units/mL and $t$ is the number of hours elapsed after the shift begins at 06:00.

Calculate the time at which the enzyme concentration drops below $50\text{ units/mL}$ . Give your answer to the nearest minute.
Calculate the exact percentage of the concentration that has been lost between 06:00 and 14:00.
Determine the instantaneous rate at which the concentration is decreasing at 06:00, providing your answer with appropriate units.
A specialized secondary analysis session begins at 11:00. Consider a function $g$ defined by:
$g(x) = K \cdot B^x$
where $g(x)$ corresponds to the enzyme concentration in units/mL $x$ hours after 11:00. Determine the value of $K$ to 1 decimal place and the value of $B$ to 2 decimal places.

An intense blizzard damages the automated supply lines of a remote Arctic monitoring station. The researchers must survive on rations and backup fuel cells. A recovery team is scheduled to reach them in exactly $120\text{ hours}$ ( $\approx 5.0\text{ days}$ ).

The specialized fuel mass consumed by the system heaters each day is given by a function $C$ , defined by:

$C(n) = 5.20e^{0.035n}$

where $n$ is the number of complete days after the supply line failure and $C(n)$ is the number of kg consumed on the $n\text{-th}$ day.

Determine the daily percentage change in fuel consumption from one day to the next.
Calculate the heating system’s fuel consumption in kilograms on the first complete day ( $n = 1$ ).

Fuel is utilized by the station cabins and slowly bleeds away due to environmental bleed-off valves. In total, the remaining fuel reserve in kilograms after the incident is given by the function:

$R(t) = 210 - 165e^{0.035t}$

where $t$ is the elapsed tracking time measured in days since the initial pipeline freeze.

Calculate the total mass of the fuel reserve in kilograms immediately after the line freeze incident occurs.
Determine whether the remaining fuel stash is a valid cause for concern during the 5-day isolation window before the rescue group arrives. Justify your answer using values calculated from the model.

A bio-chemical refinery tracks microbial populations inside an industrial fermentation vat. No calculators are allowed.

The population of active yeast cells during a temperature cycle is modeled by the function $Y$ , defined by:
$Y(t) = 300 \cdot e^{\ln(1.1) \cdot t}$
where $t$ represents the process duration in hours, with $t = 0$ as the cycle start. Determine the exact cell counts at $t = 0$ and $t = 1$ .
The population of an un-optimized competitive bacterial contaminant drops following the function $B$ , defined by:
$B(t) = 8000 \cdot 0.65^t$
where $t$ represents the process duration in hours as defined above. State whether the contamination level is ascending or descending, and calculate its exact hourly percentage change metric.

An oceanographic vessel tracks two micro-organism populations inside a marine bay enclosure over a multi-day study. No calculators are allowed.

The biomass density of a phytoplankton colony is modeled by the function $P$ , defined by:
$P(t) = 20 \cdot e^{\ln(1.4) \cdot t}$
where $t$ measures the time in days, with $t = 0$ as the initial observation baseline. Determine the exact biomass values at $t = 0$ and $t = 1$ .
The headcount density of an invasive jellyfish larva population within the same sector follows the function $J$ , defined by:
$J(t) = 3500 \cdot 0.91^t$
where $t$ is the time in days as defined above. State whether the larval population is multiplying or diminishing, and calculate its exact daily percentage rate of change.

An agricultural study measures the breakdown of a controlled organic compost heap. The data suggests that the remaining mass of the compost pile decreases by $10\%$ each week due to microbial activity.

The initial starting mass of the compost heap is $80\text{ kg}$ .

Explain why the remaining mass of the compost heap can be modeled by a function $M$ defined by $M(t) = 80 \cdot 0.9^t$ , where $t$ is the number of weeks since the start of the study and $M(t)$ is expressed in $\text{kg}$ .
Using this mathematical model, determine the exact mass of the compost heap remaining after $2\text{ weeks}$ .
Analyze the behavior of this model after a long period of time has elapsed. Will this function remain a relevant model after $100\text{ weeks}$ ? Justify your answer.

A clinical pharmacy tracks the clearance rate of an active medical compound from a patient’s body after an initial injection. Metabolic filtering models indicate that the drug concentration in the bloodstream decreases by $20\%$ each hour.

The initial concentration tracked immediately following injection is $40\text{ mg/L}$ .

Explain why the remaining medication concentration can be modeled by a function $C$ defined by $C(t) = 40 \cdot 0.8^t$ , where $t$ is the number of hours after the injection and $C(t)$ is expressed in $\text{mg/L}$ .
Using this mathematical model, determine the exact drug concentration remaining in the bloodstream after $2\text{ hours}$ .
Evaluate the limitations of this model over an extended period of time.