Problem 17 - Calculus of a Single Variable

Question

Using Properties of Logarithms In Exercises 17 and \(18,\) use the properties of logarithms to approximate the indicated logarithms, given that \(\ln 2 \approx 0.6931\) and \(\ln 3 \approx 1.0986\) \(\begin{array}{llll}{\text { (a) } \ln 6} & {\text { (b) } \ln \frac{2}{3}} & {\text { (c) } \ln 81} & {\text { (d) } \ln \sqrt{3}}\end{array}\)

Step-by-Step Solution

Answer

\(\ln 6 = 1.7917, \ln \frac{2}{3} = -0.4055, \ln 81 = 4.3944, \ln \sqrt{3} = 0.5493\).

Step 1: Approximate \(\ln 6\)

\(\ln 6\) can be written as \(\ln(2 \cdot 3)\). Using the product property it becomes \(\ln 2 + \ln 3\), which can be approximated to \(0.6931 + 1.0986 = 1.7917\).

Step 2: Approximate \(\ln \frac{2}{3}\)

\(\ln \frac{2}{3}\) can be written as \(\ln 2 - \ln 3\), using the quotient property of logarithms. This approximates as \(0.6931 - 1.0986 = -0.4055\).

Step 3: Approximate \(\ln 81\)

\(\ln 81\) can be written as \(\ln(3^4)\), using the power property it becomes \(4 \ln 3\), which approximates to \(4 \cdot 1.0986 = 4.3944\).

Step 4: Approximate \(\ln \sqrt{3}\)

\(\ln \sqrt{3}\) can be written as \(\ln \left(3^{1/2}\right)\). Using the power property of logarithms, it becomes \(\frac{1}{2} \ln 3\), which approximates as \(\frac{1}{2} \cdot 1.0986 = 0.5493\).

Key Concepts

Properties of Logarithms

Logarithms have special properties that make them a powerful tool in mathematics. They allow us to simplify complex expressions by transforming multiplication into addition, division into subtraction, and powers into products. These properties can help us solve logarithmic equations, approximate values, and simplify expressions.

• The product property states that the logarithm of a product is equal to the sum of the logarithms: \( \log_b(x \cdot y) = \log_b(x) + \log_b(y) \).


• The quotient property tells us the logarithm of a quotient is the difference of the logarithms: \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \).


• The power property allows us to bring the exponent in front of the logarithm: \( \log_b(x^n) = n \cdot \log_b(x) \).

Recognizing when and how to use these properties is essential for simplifying logarithmic expressions efficiently.

Natural Logarithm

A natural logarithm, denoted by \( \ln \), is a logarithm with base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. Natural logarithms have broad applications in science, particularly in modeling growth processes and decay phenomena, because they simplify many equations involving exponential functions.Nature frequently presents scenarios that naturally involve this base, like continuous growth in populations or interest. You'll often see \( \ln \) used in calculus, physics, and other sciences to solve problems involving rates of change and growth.

Logarithmic Approximation

Sometimes we need to estimate logarithmic values without a calculator, especially when given certain values as reference. For instance, using approximated values like \( \ln 2 \approx 0.6931 \) and \( \ln 3 \approx 1.0986 \), we can find approximate logarithmic values for larger expressions by breaking them into simpler parts using known properties of logarithms.For example, if asked to approximate \( \ln 6 \), we express it as \( \ln (2 \cdot 3) = \ln 2 + \ln 3 \). With the given approximations, this becomes \( 0.6931 + 1.0986 = 1.7917 \). Such approximations are useful for quickly solving problems in exams or during analysis without relying heavily on digital computation.

Product Property of Logarithms

The product property of logarithms allows us to transform the logarithm of a multiplication into a sum. This transformation simplifies the arithmetic, making it easier to compute or estimate logarithmic expressions.For example, considering the expression \( \ln(2 \cdot 3) \), the product property lets us rewrite it as \( \ln 2 + \ln 3 \). This is easier to handle, especially if we have approximate values for \( \ln 2 \) and \( \ln 3 \). By understanding and applying this property, we can manage complex expressions more effectively, such as when we approximate \( \ln 6 \) during calculations. This not only saves time but also provides a clearer path to the solution.