Problem 81 - College Algebra and Calculus: An Applied Approach

Question

The population of a town will double in \(t=\frac{8 \ln 3}{\ln 63-\ln 45}\) years. Find \(t\).

Step-by-Step Solution

Answer

The population will double in approximately 26.03 years.

Step 1: Simplify the logarithms

Rewrite \(\ln63-\ln45\) as \(\ln\frac{63}{45}\) as per logarithmic rules.

Step 2: Simplify the fraction

Reduce the fraction inside the logarithm to its simplest form. Hence, \(\ln\frac{63}{45}\) reduces to \(\ln\frac{7}{5}\). The expression now becomes \(t=\frac{8 \ln 3}{\ln\frac{7}{5}}\)

Step 3: Substitute values

Now that the expressions are simplified, substitute the values of the natural logarithms into the expression. Here, \(\ln3\approx1.09861\) and \(\ln\frac{7}{5}\approx0.33647\). Hence, \(t\approx\frac{8*1.09861}{0.33647}\)

Step 4: Compute \(t\)

Evaluate the expression for \(t\). Performing the calculation results in an answer of \(t\approx26.03\). Hence the population of the town will double in about 26.03 years

Key Concepts

Natural Logarithms

Natural logarithms are a fundamental mathematical concept, especially in the context of population growth and other real-world exponential processes. The natural logarithm, denoted as \(\ln x\), is essentially the power to which the constant \(\text{e}\), approximately equal to 2.71828, must be raised to obtain the value \(x\). This constant \(\text{e}\) arises naturally in various areas of mathematics, including compound interest, population dynamics, and continuous growth processes.

Understanding how to work with \(\ln x\) is crucial for solving problems involving exponential growth. In the exercise, we see \(\ln 3\) and later \(\ln \left(\frac{7}{5}\right)\). The natural logarithm allows us to translate the growth process into manageable numbers, making calculations and predictions possible. It's important to remember that the natural logarithm of a number is the inverse operation of taking \(\text{e}\) to the power of that number. Therefore, \(\ln(\text{e}^{x}) = x\) and \(\text{e}^{\ln x} = x\), which are critical relationships in logarithmic and exponential functions.

Logarithmic Properties

Logarithmic properties are the rules that govern the manipulation and simplification of logarithmic expressions. A stronger grasp on these properties allows students to confidently streamline complex logarithmic problems into more straightforward expressions.

Some of the key logarithmic properties include:

• The Product Rule: \(\ln(ab) = \ln a + \ln b\), which indicates that the logarithm of a product is equal to the sum of the logarithms of the individual factors.
• The Quotient Rule: \(\ln\left(\frac{a}{b}\right) = \ln a - \ln b\), which is used in the exercise to simplify \(\ln63 - \ln45\) into \(\ln\left(\frac{63}{45}\right)\).
• The Power Rule: \(\ln(a^b) = b\ln a\), which expresses the logarithm of a number raised to a power as the product of the exponent and the logarithm of the base number.

By applying these properties, we can transform the original expression into a more simple equation, aiding in the calculation of the population doubling time. Awareness and application of these properties are vital for solving logarithmic equations and understanding how they represent real-world situations such as population growth.

Exponential Functions

Exponential functions model situations where growth or decay is proportional to the current amount. They are ubiquitous in real-life scenarios such as population growth, radioactive decay, and interest calculations. An exponential function is defined as \(y = ab^{x}\), where \(a\) is the initial value, \(b\) is the base or the growth factor, and \(x\) represents time or the number of periods.

In the context of our population growth exercise, the doubling time is related to an exponential function where the base \(b\) would be greater than 1, indicating growth. The formula for doubling time in an exponential growth scenario can be derived using logarithms to solve for \(x\) when \(y\), the population, has doubled from its initial amount. This is where natural logarithms and their properties become invaluable, as they allow us to isolate \(x\) and solve for the time the population takes to double.

The calculation of \(t\) involves using natural logarithms to reframe the exponential growth equation into a linear form that can be manipulated algebraically. Recognizing exponential functions and their logarithmic counterparts equips students with a powerful toolset for analyzing dynamic systems that exhibit exponential behavior.