Simplifying Exponential Expressions Calculating Population Doubling Time

In order to determine the population doubling time, we are given the expression $\frac{e^{\ln 10}}{2 \ln e}$. Let's simplify this expression step by step to find the exact number of years it will take for the town's population to double. This involves understanding the properties of exponential and logarithmic functions. The key here is to recognize that the exponential function $e^x$ and the natural logarithm function $\ln x$ are inverse functions. This means that $e^{\ln x} = x$ for any positive $x$. Also, remember that the natural logarithm of $e$, denoted as $\ln e$, is equal to 1. With these properties in mind, we can simplify the numerator and the denominator of the given expression separately before combining them to get the final result. Understanding these logarithmic and exponential relationships is crucial not only for solving this problem but also for various applications in mathematics, science, and engineering where exponential growth and decay are involved. When dealing with population growth, exponential models are frequently used because populations tend to grow at a rate proportional to their current size. This leads to exponential growth, where the rate of increase becomes faster over time.

The numerator of the expression is $e^{\ln 10}$. Using the property that $e^{\ln x} = x$, we can directly simplify this to 10. This step highlights the inverse relationship between the exponential function with base $e$ and the natural logarithm. The natural logarithm, by definition, is the exponent to which $e$ must be raised to obtain a given number. Therefore, $e$ raised to the power of the natural logarithm of 10 simply returns 10. This simplification is a fundamental concept in dealing with exponential and logarithmic expressions. It allows us to convert complex expressions into simpler forms, making them easier to work with in mathematical calculations and modeling. For example, in population models, this simplification can help in determining the growth rate or the time it takes for a population to reach a certain size. In finance, it can be used to calculate the returns on investments that grow exponentially. Understanding and applying this property is essential for anyone working with exponential growth or decay phenomena.

The denominator of the expression is $2 \ln e$. Since $\ln e = 1$, the denominator simplifies to $2 \times 1 = 2$. This simplification is straightforward but important. The natural logarithm of $e$ is a fundamental constant in mathematics, much like $\pi$. Knowing that $\ln e = 1$ is essential for simplifying expressions involving natural logarithms. In this context, it helps us reduce the denominator to a simple numerical value, which is necessary for calculating the final result. This type of simplification is not just limited to mathematical exercises; it is also used extensively in various fields such as physics, engineering, and computer science. For example, in computer science, logarithms are used to analyze the efficiency of algorithms, and the base of the logarithm often plays a significant role. In physics, logarithmic scales are used to measure quantities that vary over a wide range, such as the intensity of sound (decibels) or the magnitude of earthquakes (Richter scale). Thus, understanding the properties of logarithms and their relationships with exponential functions is a cornerstone of mathematical literacy.

Now, we can simplify the entire expression $\frac{e^{\ln 10}}{2 \ln e}$ by substituting the simplified numerator and denominator. We have $\frac{10}{2}$, which simplifies to 5. Therefore, the expression $\frac{e^{\ln 10}}{2 \ln e}$ simplifies to 5. This means that the town predicts its population will double in 5 years. This result is a direct consequence of the initial simplification steps, which involved understanding the properties of exponential and logarithmic functions. The ability to simplify complex expressions like this is crucial in many areas of mathematics and its applications. For instance, in population modeling, this calculation helps us predict how long it will take for a population to double, which is an important metric for urban planning and resource management. In finance, similar calculations are used to estimate the doubling time of investments, which is a key factor in financial planning and investment strategies. The simplicity of the final answer, 5 years, underscores the power of mathematical simplification in making complex problems more manageable and understandable.

The simplified expression gives us the number of years it will take for the town's population to double. As we found, $\frac{e^{\ln 10}}{2 \ln e} = 5$. Therefore, the population of the town will double in 5 years. This is a significant piece of information for the town's planners and residents. Understanding the population doubling time allows for better planning of infrastructure, resources, and services. For example, the town can anticipate the need for more housing, schools, hospitals, and utilities as the population grows. This proactive approach is essential for sustainable development and ensuring a high quality of life for all residents. Population doubling time is also a key indicator of the town's growth rate and can be compared with regional or national averages to assess its demographic trends. Additionally, it can be used to inform decisions about economic development and job creation. A rapidly growing population may require more job opportunities, while a slower growth rate may necessitate strategies to attract new residents and businesses. Thus, the calculation of population doubling time is not just a mathematical exercise but a practical tool for informed decision-making and strategic planning at the local level.

In conclusion, by simplifying the expression $\frac{e^{\ln 10}}{2 \ln e}$, we found that the town's population is predicted to double in 5 years. This calculation demonstrates the power of understanding exponential and logarithmic functions and their applications in real-world scenarios. The ability to simplify complex expressions is a valuable skill in mathematics and beyond, allowing us to make predictions and informed decisions about various phenomena, from population growth to financial investments. In the case of this town, knowing the population doubling time is crucial for planning and development, ensuring that the town can accommodate its growing population and maintain a sustainable and thriving community. The principles used in this problem, such as the inverse relationship between exponential and logarithmic functions and the simplification of expressions, are fundamental concepts in mathematics and are applicable to a wide range of problems in science, engineering, and finance. Therefore, mastering these concepts is essential for anyone pursuing careers in these fields and for making informed decisions in everyday life. The example of population doubling time illustrates how mathematical concepts can be applied to address practical challenges and improve our understanding of the world around us.

Keywords

Exponential expression, population doubling, logarithmic functions, simplification, natural logarithm, town planning.