Problem 9 - College Algebra

Question

For the following exercises, condense to a single logarithm if possible. $$ \ln (7)+\ln (x)+\ln (y) $$

Step-by-Step Solution

Answer

The expression condenses to \( \ln(7xy) \).

Step 1: Understand the Properties of Logarithms

When combining logarithmic expressions, we use the properties of logarithms. Specifically, the property \( \ln(a) + \ln(b) = \ln(a \cdot b) \) allows us to condense the sum of logarithms into a single logarithm by converting the sum into a product inside the logarithm.

Step 2: Apply the Logarithmic Property to the First Two Terms

First, apply the property \( \ln(a) + \ln(b) = \ln(a \cdot b) \) to the first two terms: \( \ln(7) + \ln(x) = \ln(7 \cdot x) \).

Step 3: Apply the Logarithmic Property to Incorporate the Third Term

Next, use the same property with the result from Step 2 and the third term, \( \ln(y) \):\( \ln(7 \cdot x) + \ln(y) = \ln((7 \cdot x) \cdot y) \).

Step 4: Simplify the Expression

Simplify the expression to a single logarithm:\( \ln(7xy) \). This represents the expression condensed into a single logarithm.

Key Concepts

Logarithmic Expressions

Logarithmic expressions are mathematical statements that include logarithms, such as \( \ln(x) \), \( \log_{10}(x) \), or \( \log_b(x) \). These expressions allow us to represent the exponent that a base number, often 'e' or 10, must be raised to in order to obtain a given value. Understanding the nature of these expressions is key to working with them effectively.

• Common Logarithms: Logarithms with base 10. Represented as \( \log(x) \) or \( \log_{10}(x) \).
• Natural Logarithms: Logarithms with base 'e', approximately equal to 2.71828. Represented as \( \ln(x) \).
• Base Change: You can change the base of a logarithm using the change of base formula: \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \).

When dealing with logarithmic expressions, keep in mind they are not linear transformations. Instead, they pertain to exponential growth and decay processes, making them powerful tools in various mathematical and scientific disciplines.
Understanding their properties lays the groundwork for complex operations such as expanding or condensing logarithms.

Product of Logarithms

The product property of logarithms is pivotal to manipulating logarithmic expressions. This property states that the sum of the logarithms of two numbers is equal to the logarithm of the product of those numbers. Mathematically, this is expressed as:
\[ \ln(a) + \ln(b) = \ln(a \cdot b) \]
This property simplifies the process of combining multiple logarithmic terms into a single expression, enabling more straightforward computations. Here's a simple explanation:

• The expression \( \ln(7) + \ln(x) \) can be condensed by recognizing it as the logarithm of the product of its inner terms: \( \ln(7 \cdot x) \).
• By repeating this process, we further simplify our expressions. Continuing with \( \ln(y) \), we incorporate it into the existing expression: \( \ln((7 \cdot x) \cdot y) = \ln(7xy) \).

Understanding this property allows for seamless transitions from expanded to condensed forms and vice versa in logarithmic statements. This aspect of logarithms provides a base for solving many practical problems, such as those involving exponential growth scenarios like compound interest or population models.

Condensing Logarithms

Condensing logarithms involves transforming an expanded logarithmic expression into a single, compact logarithmic form. This technique is particularly useful in simplifying complex expressions and solving logarithmic equations. When we condense logarithms, we are essentially doing the reverse of expanding them.
Here’s how you can apply the condensing technique step-by-step:

• Identify the logarithmic terms that can be combined using logarithmic properties. Start with terms that share a common primary operation.
• Apply the product rule of logarithms: Combine logarithms into a product within a single logarithm if they are added together.
• Follow through with the calculation for more than two terms by sequentially applying the product rule until only one logarithm remains.

For instance, from the example \( \ln (7)+\ln (x)+\ln (y) \):
- First, pair the first two logarithms to form \( \ln(7 \cdot x) \). - Then, combine this result with the remaining logarithm, yielding \( \ln(7xy) \).
Condensing makes it easier to handle extensive calculations and interpret the log's role in equations by boiling down the expression to its essentials. This methodology amplifies clarity and efficiency when solving mathematical problems involving logarithms.