Investigation: Logarithms Revision Notes for HSC SSCE Mathematics Advanced

Investigation: Logarithms

This investigation explores logarithms through a practical application: solving a murder mystery using Newton's cooling formula. You will learn how exponential decay and logarithms can be used to determine the time of death in forensic investigations.

Learning objectives

By completing this investigation, you will be able to:

Apply logarithmic functions to solve real-world problems
Understand the heating and cooling formula
Practise identifying logarithmic data
Investigate the relationship between exponents and logarithms

The murder mystery scenario

Setting the scene

You and your classmates have rented a mansion for an end-of-year celebration. During the party, a loud scream is heard from upstairs. Everyone rushes upstairs to discover a dead body in the hallway outside the library. The deceased person is the owner of the mansion, and the identity of the killer is unknown.

Key facts about the crime

The only people present at the mansion are the students in your class, meaning the murderer must be among you. Here are the crucial pieces of evidence:

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Critical Evidence:

The mansion owner was found in the hallway near the library
Only one pathway leads from the library to the bathroom
Body temperature at $6{:}00$ p.m. was 33.89°C
Body temperature at $8{:}00$ p.m. was 30.33°C
The thermostat was set at 21.11°C

Your task is to determine what time the murder occurred and identify the killer based on who was passing through the hallway at that time.

Pre-investigation exploration

Before solving the murder mystery, consider these important questions to develop your understanding of how objects cool.

Comparing cooling rates

Think about how a corpse cools compared to everyday items:

Would a corpse cool similarly to hot coffee or tea? How might it be different?
What is a reasonable starting temperature for coffee or tea?
How long does it take for a hot drink to cool down?
At what temperature would you consider coffee or tea to be cold?
Would the same temperature be considered cold for a human body?

Modelling temperature change

When graphing temperature change over time:

Plot temperature on the $y$ -axis and time on the $x$ -axis
Consider whether a linear graph would accurately represent cooling
What type of graph best models this situation? Why?

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Key Concept: Exponential Cooling

The answer is that cooling follows an exponential pattern, not a linear one. The temperature decreases rapidly at first, then slows down as it approaches room temperature.

Optional experimental investigation

If you have access to a cooking thermometer, you can test these ideas:

Note the room temperature
Make a cup of hot coffee or tea
Measure the temperature at regular intervals:
- Immediately after making it
- At $2$ , $4$ , $5$ , $10$ , $15$ , $20$ , $30$ , and $60$ minutes
Create a graph with time on the $x$ -axis and temperature on the $y$ -axis
Observe the shape of the curve
Predict what the temperature would be after $2$ or $3$ hours

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This hands-on experiment will help you visualize the exponential cooling pattern and understand how temperature changes over time in a real-world context.

Newton's cooling formula

Understanding the formula

The cooling of a body follows an exponential decay pattern described by Newton's formula for cooling:

$T(t) = T_{env} + (T_0 - T_{env})e^{-kt}$

Where:

$T(t)$ is the temperature of the body at time $t$
$T_{env}$ is the temperature of the surrounding environment
$T_0$ is the initial temperature of the body (average living body temperature is approximately $37°\text{C}$ )
$t$ is the time after death in hours
$k$ is a constant that can be calculated using the formula below

Calculating the constant $k$

The constant $k$ depends on the specific situation and can be found using:

$e^{-2k} = \frac{T(t+2) - T_{env}}{T(t) - T_{env}}$

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This equation uses two temperature measurements taken 2 hours apart. This is why we need both temperature readings from the crime scene.

Connection to logarithms

An exponential equation such as $e^x = 3$ can be rewritten as the equivalent logarithmic equation $x = \log_e 3$ .

We often use the special notation $\ln$ for the natural logarithm (base $e$ ) instead of $\log_e$ .

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Logarithmic Relationship:

Therefore: $x = \log_e 3 = \ln 3$

Logarithms are the inverse of exponential functions, which allows us to solve for unknown exponents.

Solving the murder mystery

Now let's apply Newton's cooling formula to determine when the murder occurred. This section demonstrates the complete problem-solving process step by step.

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Worked Example: Determining Time of Death

Let $t$ represent the time (in hours) after death when the body was found.

Given information:

At time $t$ (when the body was found): temperature = $33.89°\text{C}$
At time $t + 2$ (two hours later): temperature = $30.33°\text{C}$
Environmental temperature: $T_{env} = 21.11°\text{C}$
Average living body temperature: $T_0 = 37°\text{C}$

Step 1: Calculate the constant $k$

Using the formula for $k$ :

$e^{-2k} = \frac{T(t+2) - T_{env}}{T(t) - T_{env}}$

Substituting the known values:

$e^{-2k} = \frac{30.33 - 21.11}{33.89 - 21.11}$

$e^{-2k} = \frac{9.22}{12.78}$

$e^{-2k} = 0.7215...$

Step 2: Solve for $k$ using logarithms

Taking the natural logarithm of both sides:

$-2k = \ln(0.7215...)$

$-2k = -0.3262...$

$k = 0.1631...$

Step 3: Find the time $t$

Using Newton's cooling formula with the first temperature reading:

$33.89 = 21.11 + (37 - 21.11)e^{-kt}$

$33.89 = 21.11 + 15.89e^{-0.1631t}$

$12.78 = 15.89e^{-0.1631t}$

$\frac{12.78}{15.89} = e^{-0.1631t}$

$0.8044... = e^{-0.1631t}$

Taking the natural logarithm of both sides:

$\ln(0.8044...) = -0.1631t$

$-0.2177... = -0.1631t$

$t = 1.335... \text{ hours}$

Step 4: Convert to hours and minutes

$t \approx 1$ hour $20$ minutes

Step 5: Determine the time of death

Since the body was found at $6{:}00$ p.m., the murder occurred approximately $1$ hour $20$ minutes earlier.

Time of death: approximately 4:40 p.m.

Identifying the murderer

Based on the time cards showing when each person passed through the hallway:

Time cards: $4{:}00$ , $4{:}10$ , $4{:}20$ , $4{:}30$ , $4{:}40$ , $4{:}50$ , $5{:}00$ , $5{:}10$ , $5{:}20$ p.m.

chatImportant

The person (or pair) who drew the 4:40 p.m. time card is most likely the murderer, as this matches the calculated time of death.

Real-world applications

Logarithms are not just abstract mathematical concepts - they have important real-world applications in forensic science. When a person dies, their body temperature begins to decrease to match the surrounding environment. This cooling process does not occur at a constant rate; it is exponential, meaning the temperature drops quickly at first and then slows down over time.

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Forensic Applications:

Forensic scientists use Newton's cooling formula and logarithms to estimate the time of death in murder investigations. By measuring the body temperature at two different times and knowing the environmental temperature, they can work backwards to determine when the person died.

This technique is particularly valuable in the first 24 hours after death, when body temperature is still significantly different from the environmental temperature.

Key considerations

When using the heating and cooling formula, remember these important factors that affect the accuracy of time-of-death calculations:

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Key Points to Remember:

The cooling process is exponential, not linear
Different objects cool at different rates (the constant $k$ varies)
Environmental temperature affects the cooling rate
The formula assumes the environmental temperature remains constant
Initial temperature and environmental temperature are crucial values

Summary

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Essential Concepts:

Newton's cooling formula describes exponential temperature decay: $T(t) = T_{env} + (T_0 - T_{env})e^{-kt}$
Logarithms are the inverse of exponential functions, allowing us to solve for time
The natural logarithm $\ln$ is used with base $e$ in exponential growth and decay problems
Real-world applications of logarithms include forensic science, particularly determining time of death
Cooling is exponential - objects cool quickly at first, then more slowly as they approach environmental temperature

Investigation: Logarithms (HSC SSCE Mathematics Advanced): Revision Notes