MTH120 College Algebra Chapter 6.7

 6.7 Exponential and Logarithmic Models

Exponential and logarithmic models are powerful tools for describing various real-world phenomena that involve growth, decay, or proportional relationships. These models are frequently used in fields such as finance, economics, biology, physics, and engineering to make predictions and analyze data. Here, we'll explore some key concepts related to exponential and logarithmic models:

Exponential Models:

1. Exponential Growth: Exponential growth models describe situations in which a quantity increases rapidly over time. The general form of an exponential growth model is given by $�(�)={�}_0\cdot {�}^{��}$, where:

   • $�(�)$ is the quantity at time $�$.
   • ${�}_0$ is the initial quantity.
   • $�$ is the growth rate (as a decimal).
   • $�$ is time.

2. Exponential Decay: Exponential decay models describe situations in which a quantity decreases rapidly over time. The general form is similar to exponential growth but with a negative exponent: $�(�)={�}_0\cdot {�}^{-��}$.

3. Doubling Time and Half-Life: In exponential growth, the doubling time is the time it takes for the quantity to double. In exponential decay, the half-life is the time it takes for the quantity to decrease by half.

Logarithmic Models:

1. Logarithmic Growth: Logarithmic growth models describe situations in which a quantity grows at a decreasing rate, eventually approaching a limit or carrying capacity. The general form is $�(�)={�}_{\text{max}}\mathrm{/}(1+�{�}^{-��})$, where:

   • $�(�)$ is the quantity at time $�$.
   • ${�}_{\text{max}}$ is the maximum attainable quantity.
   • $�$ and $�$ are constants.

2. Logarithmic Decay: Logarithmic decay models describe situations in which a quantity decreases at a decreasing rate. The general form is similar to logarithmic growth.

3. Logistic Growth: Logistic growth models are a type of logarithmic growth model that includes a carrying capacity. They are often used to describe population growth.

Applications:

1. Finance: Compound interest and continuous compounding are modeled using exponential functions.

2. Biology: Population growth, bacterial growth, and the spread of diseases can be modeled using exponential and logarithmic functions.

3. Physics: Radioactive decay, heat transfer, and wave attenuation are examples of physical processes modeled using these functions.

4. Economics: Economic growth, depreciation of assets, and consumer demand are often described by exponential and logarithmic models.

5. Engineering: Engineering applications include modeling electrical circuits, fluid flow, and structural stability.

6. Environmental Science: Models for deforestation, climate change, and ecological balance often involve exponential and logarithmic equations.

Understanding exponential and logarithmic models is crucial for making predictions, analyzing data, and making informed decisions in various fields. These models provide valuable insights into how quantities change over time and how different factors impact growth or decay.

Exponential growth and decay are fundamental concepts in various fields, and they can be modeled using mathematical equations. Here are examples of exponential growth and decay models:

Exponential Growth:

1. Population Growth: The population of a species can be modeled using exponential growth. For example, if a population of bacteria doubles every hour, you can use the formula $�(�)={�}_0\cdot 2^{(�\mathrm{/}ℎ)}$, where:

   • $�(�)$ is the population at time $�$.
   • ${�}_0$ is the initial population.
   • $�$ is the time.
   • $ℎ$ is the doubling time.

   If you start with 100 bacteria (${�}_0=100$) and the doubling time is 1 hour ($ℎ=1$), after 5 hours, you'd have $�(5)=100\cdot 2^{(5\mathrm{/}1)}=3200$ bacteria.

2. Financial Investments: Compound interest in financial investments is an example of exponential growth. The formula for compound interest is $�(�)={�}_0\cdot (1+�\mathrm{/}�{)}^{��}$, where:

   • $�(�)$ is the amount of money at time $�$.
   • ${�}_0$ is the initial principal.
   • $�$ is the annual interest rate (as a decimal).
   • $�$ is the number of times interest is compounded per year.
   • $�$ is the time in years.

   If you invest $1,000 (${�}_0=1000$) at an annual interest rate of 5% ($�=0.05$) compounded quarterly ($�=4$), after 10 years, you'd have (A(10) = 1000 \cdot (1 + 0.05/4)^{4 \cdot 10} \approx $1,628.89.

Exponential Decay:

1. Radioactive Decay: The decay of radioactive isotopes follows exponential decay. The formula for radioactive decay is $�(�)={�}_0\cdot {�}^{-��}$, where:

   • $�(�)$ is the remaining quantity at time $�$.
   • ${�}_0$ is the initial quantity.
   • $�$ is the decay constant.
   • $�$ is the time.

   If you start with 100 grams of a radioactive substance (${�}_0=100$) with a decay constant of 0.01 per year ($�=0.01$), after 5 years, you'd have $�(5)=100\cdot {�}^{-(0.01\cdot 5)}\approx 60.65$ grams remaining.

2. Temperature Cooling: The cooling of an object follows exponential decay. The formula for temperature cooling is $�(�)={�}_0\cdot {�}^{-��}$, where:

   • $�(�)$ is the temperature at time $�$.
   • ${�}_0$ is the initial temperature.
   • $�$ is the cooling constant.
   • $�$ is the time.

   If a cup of coffee cools from 90°C (${�}_0=90$) to room temperature with a cooling constant of 0.02 per minute ($�=0.02$), after 10 minutes, the temperature will be (T(10) = 90 \cdot e^{-(0.02 \cdot 10)} \approx 73.80°C.

These examples illustrate how exponential growth and decay models are applied in various real-world scenarios. Understanding these models is essential for making predictions, solving problems, and analyzing data in fields such as biology, finance, physics, and environmental science.

The exponential function $�={�}_0{�}^{��}$ has several key characteristics:

1. Base: The base of the exponential function is $�$, which is Euler's number, approximately equal to 2.71828. This base is fundamental in calculus and often used in exponential growth and decay models.

2. Initial Value: ${�}_0$ represents the initial value or the value of the function at $�=0$. It determines the starting point of the exponential growth or decay.

3. Growth or Decay: The value of $�$ determines whether the function represents exponential growth or decay. If $�>0$, the function models exponential growth, while if $�<0$, it models exponential decay.

4. Rate of Growth/Decay: The absolute value of $�$ determines the rate of growth or decay. A larger absolute value of $�$ results in a faster rate of change.

5. Time: $�$ represents time or the independent variable. It is the variable with respect to which the function's value changes.

6. Continuous Change: The exponential function is continuous, which means it doesn't have discrete jumps or steps. The value of the function changes continuously as time progresses.

7. Asymptote: Exponential functions have horizontal asymptotes, which are lines that the graph approaches but never crosses. The asymptote is determined by the initial value ${�}_0$. If $�>0$, the asymptote is at $�=0$, while if $�<0$, the asymptote is at $�=0$.

8. Monotonicity: Exponential functions are monotonic, meaning they are always increasing (for positive $�$) or decreasing (for negative $�$) as $�$ increases. They do not change direction.

9. Unbounded Growth/Decay: Exponential functions can grow or decay without bound, meaning there's no limit to how large or small the values can become. However, the rate of growth or decay may vary.

10. Real-World Applications: Exponential functions are commonly used to model processes such as population growth, compound interest in finance, radioactive decay, and the spread of diseases.

Understanding these characteristics is crucial when working with exponential functions in various applications, from finance to science and engineering. These functions offer a powerful way to describe and predict continuous growth and decay in the real world.

The half-life formula is used to describe the time it takes for a substance to decay or reduce by half. It's commonly used in various fields, especially in nuclear physics, chemistry, and medicine. You can find more formulas like this if your taking nuclear technician classes. The general half-life formula is:

${�}_{\text{half}}=\frac{\ln (2)}{�}$

Where:

• ${�}_{\text{half}}$ is the half-life of the substance.
• $�$ is the decay constant.

Here are some examples of half-life calculations:

1. Radioactive Decay:

• A sample of a radioactive isotope has a decay constant $�=0.1{\text{ yr}}^{-1}$.
• Calculate the half-life of the isotope.

Using the half-life formula: ${�}_{\text{half}}=\frac{\ln (2)}{0.1}\approx \frac{0.6931}{0.1}\approx 6.931\text{ years}$

So, the half-life of the radioactive isotope is approximately 6.931 years.

2. Medication Half-Life:

• A drug with a decay constant $�=0.05{\text{ hr}}^{-1}$ is administered to a patient.
• Calculate the half-life of the drug.

Using the half-life formula: ${�}_{\text{half}}=\frac{\ln (2)}{0.05}\approx \frac{0.6931}{0.05}\approx 13.862\text{ hours}$

The half-life of the drug is approximately 13.862 hours.

3. Carbon-14 Dating:

• Carbon-14 (${}^{14}\text{C}$) is used for radiocarbon dating, and it has a known decay constant.
• The decay constant for carbon-14 is $�=1.21\times 10^{-4}{\text{ yr}}^{-1}$.
• Calculate the half-life of carbon-14.

Using the half-life formula: ${�}_{\text{half}}=\frac{\ln (2)}{1.21\times 10^{-4}}\approx \frac{0.6931}{1.21\times 10^{-4}}\approx 5730\text{ years}$

The half-life of carbon-14 is approximately 5730 years.

These examples illustrate how the half-life formula can be applied to various situations involving decay processes. The half-life is a valuable concept for understanding how long it takes for a substance to decrease to half of its original amount, and it plays a significant role in scientific and medical research.

The function that describes radioactive decay is typically an exponential decay function. Radioactive decay refers to the process by which the number of radioactive atoms in a sample decreases over time. The general form of the exponential decay function is:

$�(�)={�}_0\cdot {�}^{-��}$

Where:

• $�(�)$ is the remaining quantity of radioactive material at time $�$.
• ${�}_0$ is the initial quantity of radioactive material at $�=0$.
• $�$ is the decay constant.
• $�$ is the time.

The specific form of the function may vary based on the radioactive isotope being considered and its decay characteristics. Here's how you can determine the function that describes radioactive decay for a given isotope:

1. Half-Life: First, you need to know the half-life (${�}_{\text{half}}$) of the radioactive isotope. The half-life is the time it takes for half of the radioactive material to decay.

2. Decay Constant ($�$): Calculate the decay constant ($�$) using the formula: $�=\frac{\ln (2)}{{�}_{\text{half}}}$

3. Form of the Function: The function that describes radioactive decay for the specific isotope is of the form $�(�)={�}_0\cdot {�}^{-��}$ where ${�}_0$ is the initial quantity at $�=0$, and $�$ is the decay constant you calculated.

4. Application: You can use this function to predict the remaining quantity of the radioactive material at any given time $�$.

For example, if you're dealing with a radioactive isotope with a half-life of 10 years and you start with 100 grams of the material, the function describing its decay would be:

$�(�)=100\cdot {�}^{-\frac{\ln (2)}{10}�}$

This function allows you to determine how much of the substance remains at any point in time.

Different radioactive isotopes have different half-lives, and the specific decay constant ($�$) will vary accordingly. Understanding the half-life and the associated decay constant is crucial for modeling and predicting the behavior of radioactive materials in various applications, including radiocarbon dating, nuclear physics, and medical imaging.

Radiocarbon dating is a commonly used method for determining the age of an object containing organic material by measuring the decay of the radioactive isotope carbon-14 (${}^{14}\text{C}$). Here's how radiocarbon dating works, along with some examples:

Principle of Radiocarbon Dating:

• Radiocarbon dating is based on the principle that all living organisms contain a small but measurable amount of radioactive carbon-14. When an organism dies, it stops acquiring new carbon-14, and the existing carbon-14 in its tissues begins to decay.

Half-Life of Carbon-14:

• Carbon-14 has a known half-life of approximately 5,730 years. This means that after 5,730 years, half of the carbon-14 in a sample will have decayed.

The Radiocarbon Dating Process:

1. A sample containing organic material (e.g., wood, bone, cloth) is collected from an archaeological site or another source.

2. The sample is prepared, typically by converting it into carbon dioxide gas (CO2).

3. The carbon dioxide gas is then subjected to a technique called accelerator mass spectrometry (AMS). This process counts the carbon-14 atoms in the sample and compares them to the stable carbon-12 (${}^{12}\text{C}$ and carbon-13 (${}^{13}\text{C}$) isotopes.

4. By measuring the ratio of carbon-14 to carbon-12 or carbon-13, the age of the sample can be calculated using the known half-life of carbon-14.

Radiocarbon Dating Examples:

1. Archaeological Findings:

   • Example: A piece of wood is found at an archaeological site.
   • Radiocarbon dating determines that the wood is 2,000 years old.
   • This means the wood came from a tree that lived approximately 2,000 years ago.

2. Historical Documents:

   • Example: A piece of parchment from an ancient book is tested.
   • Radiocarbon dating reveals that the parchment is 500 years old.
   • This confirms the age of the book or document.

3. Carbon-14 in Living Organisms:

   • Example: A study of a contemporary animal species, such as an elephant or a sea turtle, is conducted.
   • Radiocarbon dating shows that the carbon-14 ratio in these animals is consistent with the known atmospheric levels of carbon-14.
   • This provides a baseline for comparing carbon-14 levels in ancient organisms.

4. Climate Science:

   • Example: Carbon-14 dating of tree rings (dendrochronology) is used to study climate change.
   • By dating tree rings from different periods, researchers can create a timeline of climate variations.

Radiocarbon dating is a valuable tool for dating materials from the past up to about 50,000 years ago. It has revolutionized the field of archaeology, anthropology, and other sciences by providing a reliable method for dating artifacts, fossils, and other organic remains. However, it has limitations, such as the inability to date materials older than about 50,000 years, as carbon-14 decays to undetectable levels. For older materials, other dating methods like uranium-series dating or potassium-argon dating are used.

To determine the age of an object based on the percentage of carbon-14 remaining, you can use the concept of radioactive decay and the half-life of carbon-14. The formula for calculating the remaining fraction of carbon-14 in an object is:

$�(�)={�}_0\cdot {\left(\frac{1}{2}\right)}^{\frac{�}{{�}_{\text{half}}}}$

Where:

• $�(�)$ is the remaining fraction of carbon-14 at time $�$.
• ${�}_0$ is the initial fraction of carbon-14.
• $�$ is the time that has passed.
• ${�}_{\text{half}}$ is the half-life of carbon-14, which is approximately 5,730 years.

If you're given the percentage of carbon-14 remaining, you can convert it into a fraction by dividing by 100. For example, if you're told that an object contains 25% of the original carbon-14, you would use ${�}_0=0.25$ in the formula. Then, you can solve for the age ($�$).

Let's use an example:

Example: You have a piece of wood that contains 15% of the original carbon-14. How old is the wood?

1. Convert the percentage to a fraction: ${�}_0=0.15$.

2. Use the half-life of carbon-14: ${�}_{\text{half}}=5,730$ years.

3. Plug these values into the formula: $0.15={\left(\frac{1}{2}\right)}^{\frac{�}{5730}}$

4. To solve for $�$, take the natural logarithm of both sides to get rid of the exponent: $\ln (0.15)=\frac{�}{5730}$

5. Multiply both sides by 5730 to solve for $�$: $�=5730\cdot \ln (0.15)\approx 3,207\text{ years}$

So, the wood is approximately 3,207 years old based on the remaining fraction of carbon-14. This calculation assumes that the amount of carbon-14 in the atmosphere has remained relatively constant over time, which is a reasonable assumption for dating objects up to about 50,000 years old.

The doubling time is the time it takes for a quantity to double in size through exponential growth. To calculate the doubling time, you can use the following formula:

${�}_{\text{double}}=\frac{\ln (2)}{�}$

Where:

• ${�}_{\text{double}}$ is the doubling time.
• $�$ is the growth rate (expressed as a fraction).

Here's how to calculate the doubling time:

1. Determine the growth rate ($�$) by dividing the natural logarithm of the final quantity (${�}_{�}$) by the time it took to reach that quantity ($�$): $�=\frac{\ln ({�}_{�})-\ln ({�}_0)}{�}$

2. Once you have the growth rate ($�$), you can calculate the doubling time using the formula: ${�}_{\text{double}}=\frac{\ln (2)}{�}$

Here's an example:

Example: A population of rabbits started with 100 individuals, and after 3 years, it grew to 400 individuals. Calculate the doubling time of the rabbit population.

1. Calculate the growth rate ($�$) using the population size at the start (${�}_0$), the population size after a certain time (${�}_{�}$), and the time ($�$): $�=\frac{\ln (400)-\ln (100)}{3}=\frac{\ln (4)-\ln (1)}{3}=\frac{\ln (4)}{3}$

2. Calculate the doubling time (${�}_{\text{double}}$) using the formula: ${�}_{\text{double}}=\frac{\ln (2)}{�}=\frac{\ln (2)}{\frac{\ln (4)}{3}}$

   Using the properties of logarithms (logarithm of a number to a power): ${�}_{\text{double}}=\frac{3\ln (2)}{\ln (4)}$

Now, calculate ${�}_{\text{double}}$:

${�}_{\text{double}}\approx \frac{3\cdot 0.6931}{1.3863}\approx 1.5\text{ years}$

So, the doubling time of the rabbit population is approximately 1.5 years. This means that, on average, it takes 1.5 years for the population to double in size under the given growth rate.

Newton's Law of Cooling is a mathematical model used to describe the cooling or heating of an object over time. It states that the rate of change of the temperature of an object is directly proportional to the difference between the object's temperature and the ambient temperature. The law can be expressed with the following differential equation:

$\frac{��}{��}=-�(�-{�}_{�})$

Where:

• $\frac{��}{��}$ is the rate of change of temperature with respect to time.
• $�$ is the temperature of the object at time $�$.
• ${�}_{�}$ is the ambient temperature.
• $�$ is the cooling or heating constant.

Solving this differential equation allows us to predict how the temperature of the object changes over time.

Here's how to use Newton's Law of Cooling in a practical example:

Example: Cooling Coffee

Imagine you have a cup of hot coffee at a temperature of 90°C, and the room temperature (ambient temperature) is 20°C. The coffee is placed on a table, and you want to know how long it will take for the coffee to cool down to 50°C.

1. Define the parameters:

   • $�(0)=90$, which is the initial temperature of the coffee.
   • ${�}_{�}=20$, which is the ambient temperature.
   • The cooling constant $�$ would depend on the specific conditions (insulation, type of cup, etc.).

2. Formulate the differential equation: $\frac{��}{��}=-�(�-{�}_{�})$

3. To find $�$, you need to know the rate of change of temperature at some point. For example, if you measure the temperature to be 70°C after 10 minutes, you can use this information to calculate $�$. The equation becomes: $\frac{��}{��}=-�(70-20)$

4. Solve the differential equation for $�$.

5. Once you have $�$, you can integrate the differential equation to find the time ($�$) it takes for the coffee to cool to 50°C: $\int \frac{��}{�-{�}_{�}}=-�\int ��$

6. Plug in the values and solve the integral.

This will give you an equation that represents the time it takes for the coffee to cool to 50°C based on the initial conditions and cooling constant. Solving this equation will provide you with the time it takes for the coffee to reach the desired temperature.

It's important to note that $�$ depends on various factors such as the material of the cup, air circulation, and more. Therefore, in practice, you might need to perform experiments or consult data tables to determine a suitable value for $�$ in specific situations.

Logistic growth models are used to describe the growth of populations or the spread of phenomena when there are limitations to growth. These models are particularly relevant in biology and ecology, as they take into account factors such as carrying capacity and environmental constraints. The logistic growth model is typically expressed as follows:

$�(�)=\frac{�}{1+\left(\frac{�-{�}_0}{{�}_0}\right){�}^{-��}}$

Where:

• $�(�)$ is the population or quantity of interest at time $�$.
• $�$ is the carrying capacity, which represents the maximum population or quantity that the environment can support.
• ${�}_0$ is the initial population or quantity at $�=0$.
• $�$ is the growth rate.
• $�$ is time.

To use the logistic growth model, you typically have to know or estimate the values of $�$, ${�}_0$, and $�$ based on data and conditions relevant to your specific application. Here's how to use the logistic growth model in a practical example:

Example: Modeling the Growth of a Fish Population

Suppose you are studying a population of fish in a pond. You have data indicating that the initial population (${�}_0$) was 100 fish, the carrying capacity ($�$) of the pond is 500 fish, and the growth rate ($�$) is 0.2 per year.

1. Formulate the logistic growth model with the provided values: $�(�)=\frac{500}{1+\left(\frac{500-100}{100}\right){�}^{-0.2�}}$

2. Use this model to make predictions. For instance, you might want to know how many fish are expected to be in the pond after 5 years ($�=5$).

   $�(5)=\frac{500}{1+\left(\frac{500-100}{100}\right){�}^{-0.2\cdot 5}}$

3. Calculate $�(5)$ to find the expected population of fish after 5 years.

In this example, you can also use the logistic growth model to make predictions about population behavior over time, assess the impact of different initial conditions or growth rates, and analyze the effects of environmental factors on population growth. Logistic growth models are valuable tools in ecology, epidemiology, and other fields where growth is constrained by limited resources or space.

Choosing an appropriate model for data is a critical step in statistical analysis and modeling. The choice of a model depends on the nature of the data, the research question, and the underlying assumptions. Here's a general guideline for selecting an appropriate model for your data:

1. Understand Your Data:

   • Begin by thoroughly understanding your data. Consider the type of data (continuous, discrete, categorical), its distribution, and any patterns or trends.

2. Define Your Research Question:

   • Clearly state your research question and objectives. What are you trying to learn or predict from the data?

3. Visualize the Data:

   • Create data visualizations, such as histograms, scatterplots, or box plots, to gain insights into the data's distribution and relationships.

4. Consider Data Transformation:

   • If the data does not meet the assumptions of the model you're considering, think about data transformations to make it more suitable for modeling. Common transformations include logarithmic, square root, or Box-Cox transformations.

5. Choose the Right Model Type:

   • Select a model type that aligns with your research question:
     • Linear Models: Use linear regression for modeling relationships between variables when the relationship appears to be linear.
     • Nonlinear Models: Consider polynomial regression, exponential, logistic, or other nonlinear models when the relationship between variables is nonlinear.
     • Time Series Models: If your data is collected over time, consider time series models, such as ARIMA, for forecasting and analyzing trends.
     • Classification Models: Use classification models (e.g., logistic regression, decision trees, random forests) for predicting categorical outcomes.
     • Clustering Models: If you want to group similar data points, consider clustering algorithms like k-means or hierarchical clustering.
     • Dimensionality Reduction Models: When working with high-dimensional data, dimensionality reduction techniques like principal component analysis (PCA) or t-SNE can be useful.
     • Survival Analysis Models: For data involving time-to-event outcomes, survival analysis models like Kaplan-Meier or Cox regression are appropriate.

6. Consider Model Assumptions:

   • Be aware of the assumptions of the chosen model. Linear regression, for example, assumes linearity, independence of errors, and homoscedasticity.

7. Evaluate Model Fit:

   • Fit the model to your data and assess its performance using appropriate metrics (e.g., R-squared for linear regression, AIC for model selection). Cross-validation can help assess the model's generalization performance.

8. Iterate and Refine:

   • It's often necessary to iterate and refine your model choice and approach based on the initial results and model diagnostics.

9. Consult Domain Experts:

   • If available, consult domain experts who can provide valuable insights into the data and assist in model selection.

10. Consider Ensembling:

    • In some cases, combining multiple models through ensemble techniques like bagging or boosting can improve predictive performance.

Remember that selecting the right model is often an iterative process, and it may require experimentation and evaluation. Additionally, the choice of the appropriate model can impact the quality and reliability of your results, so it's essential to choose wisely.

To express an exponential model with a different base (other than e) in base e (natural logarithm), you can use the following conversion formula:

If you have an exponential model of the form $�=�\cdot {�}^{�}$, and you want to express it in base e (natural logarithm), you can rewrite it as:

$�=�\cdot {�}^{�}=�\cdot {�}^{\ln ({�}^{�})}$

Now, using the properties of logarithms, you can move the exponent $�$ in front of the natural logarithm:

$�=�\cdot {�}^{�\cdot \ln (�)}$

This expression represents the same exponential growth or decay model as the original, but it's expressed in base e. The parameter $\ln (�)$ is essentially the rate at which the quantity grows or decays.

For example, if you have an exponential model in base 10, such as $�=100\cdot 10^{�}$, and you want to express it in base e:

$�=100\cdot {�}^{\ln (10^{�})}$

Using the properties of logarithms:

$�=100\cdot {�}^{�\cdot \ln (10)}$

So, the equivalent model in base e is $�=100\cdot {�}^{�\cdot 2.3026}$, where 2.3026 is the natural logarithm of 10.

This conversion is particularly useful when working with exponential models in different bases and when you want to take advantage of the properties of the natural logarithm in mathematical analysis or modeling.

1. Exponential Decay of Medication in the Body:

   • Example: Elimination of a drug from the body.
   • Formula: $�(�)={�}_0{�}^{-��}$
   • Calculation: If a drug with an initial concentration of 200 mg (${�}_0=200$) has a half-life of 4 hours (${�}_{\text{half}}=4$), the concentration after 8 hours ($�=8$) would be $�(8)=200\cdot {�}^{-\frac{\ln (2)}{4}\cdot 8}\approx 25��$.

2. Decay of Light Intensity in Optics:

   • Example: Light intensity through a filter or medium.
   • Formula: $�(�)={�}_0{�}^{-��}$
   • Calculation: If light passes through a medium that reduces its intensity by 20% per minute ($�=0.20$), the intensity after 5 minutes ($�=5$) would be $�(5)={�}_0{�}^{-0.20\cdot 5}\approx 0.409{�}_0$.