Problem 25 - Calculus Single Variable

Question

In Exercises \(23-26,\) use an identity to simplify the sum. $$ \sum_{j=2}^{6} \ln (j) $$

Step-by-Step Solution

Answer

The simplified form of the sum is \( \ln(720) \).

Step 1: Review the sum expression

The given expression is \( \sum_{j=2}^{6} \ln(j) \). This represents the sum of the natural logarithms of integers from 2 to 6 inclusive.

Step 2: Apply the logarithm identity

Use the property of logarithms that states \( \ln(a) + \ln(b) = \ln(ab) \). This means we can combine all the terms under a single logarithm: \( \sum_{j=2}^{6} \ln(j) = \ln(2 \times 3 \times 4 \times 5 \times 6) \).

Step 3: Calculate the product inside the logarithm

Compute the product of the numbers: \( 2 \times 3 \times 4 \times 5 \times 6 = 720 \).

Step 4: Simplify the logarithmic expression

Substitute the product back into the logarithmic expression: \( \ln(2 \times 3 \times 4 \times 5 \times 6) = \ln(720) \). Therefore, the simplified form of the sum is \( \ln(720) \).

Key Concepts

Logarithm Properties

Logarithms have a special set of properties that makes them very useful in mathematical calculations. One key property is the product rule for logarithms:

• When you add two logarithms together, you are essentially multiplying their inputs: \( \ln(a) + \ln(b) = \ln(ab) \)

This property allows us to transform a sum into a single logarithm, making complex expressions more manageable.
Another important property is the quotient rule, which states that the difference of two logarithms can be expressed as the logarithm of their quotient:

• \( \ln(a) - \ln(b) = \ln \left( \frac{a}{b} \right) \)

These rules help in simplifying logarithmic expressions and are fundamental when dealing with equations involving logarithms.

Natural Logarithm

The natural logarithm, often written as \( \ln(x) \), is a special type of logarithm. It uses the base \( e \), where \( e \approx 2.71828 \), and is considered one of the most important numbers in mathematics.Natural logarithms are widely used in calculus, science, and engineering because they have nice mathematical properties, which simplify differentiation and integration.
For instance, the derivative of \( \ln(x) \) is \( \frac{1}{x} \), and its integral, \( \int \ln(x) \, dx \), leads to expressions that are easier to handle than logarithms with other bases.Understanding \( \ln \) is crucial for delving deeper into logarithmic operations, particularly when working with exponential functions, given the unique relationship between \( e^x \) and \( \ln(x) \).

Summation

Summation is a concise way to express the addition of a sequence of numbers. It's often seen in mathematical and scientific contexts.The notation \( \sum_{j=2}^{6} \ln(j) \) is a summation and indicates that you should add together the natural logarithms of numbers starting from 2 and ending at 6.
Summations are neat tools for expressing long sequences in a compact form, preventing errors and simplifying calculations.They are particularly useful when combined with logarithm properties, as seen in the exercise. By converting the sum into a product under a single logarithm, the process becomes more streamlined, and calculations are minimized.

Product

The product refers to the result of multiplying a sequence of numbers together. In the context of logarithmic summation, this operation plays a vital role.Using the example in our exercise, \( 2 \times 3 \times 4 \times 5 \times 6 \) results in 720. This step converts the sum of multiple logarithms into the logarithm of a single value, \( \ln(720) \), thanks to the logarithm property of converting sums into products.
This conversion is particularly beneficial because it enables easy handling of large expressions and makes solving problems more straightforward.Understanding how to compute products quickly and accurately is essential in various fields, particularly in simplifying complex arithmetic operations.