Problem 62 - College Algebra

Question

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 4 \ln x+7 \ln y-3 \ln z $$

Step-by-Step Solution

Answer

The condensed expression is \(\ln (\frac{x^4*y^7}{z^3})\)

Step 1: Use property of logarithms

First let's use the logarithm property \(a\log_b(x) = \log_b(x^a)\) to simplify each term: \n\n\(4 \ln x = \ln x^4\)\n\n\(7 \ln y = \ln y^7\)\n\n\(3 \ln z = \ln z^3\)\n\nSo, the expression becomes \(\ln x^4 + \ln y^7 - \ln z^3\)

Step 2: Apply product and quotient rules

Now we can apply the product and quotient rules. The product rule \(\log_b(x) + \log_b(y) = \log_b(x*y)\) applies to \(\ln x^4 + \ln y^7\) to produce \(\ln (x^4*y^7)\). The quotient rule \(\log_b(x) - \log_b(y) = \log_b(\frac{x}{y})\) applies to the result and \(\ln z^3\) to produce \(\ln (\frac{x^4*y^7}{z^3})\)

Step 3: Simplify the answer

Finally, simplify the answer using the properties of products and quotients to get the final result: \(\ln (\frac{x^4*y^7}{z^3})\)

Key Concepts

Logarithmic Expressions

A logarithmic expression involves logarithms, which are the inverse operations of exponentiation. In simple terms, they answer the question: "To what power must a specific base be raised to obtain a given number?" For example, if you have \(\log_b(a) = c\), this means \(b^c = a\). Logarithmic expressions often appear in mathematical problems involving exponential growth or decay, probability, or calculations requiring multiplication and division to be simplified using addition and subtraction.

In our case, the given expression \(4 \ln x + 7 \ln y - 3 \ln z\) is a combination of logarithmic terms, each including the natural logarithm, abbreviated as \ln\. The natural logarithm uses the number \(e\) (approximately 2.71828) as its base. Simplifying logarithmic expressions like this one can often help us see relationships and simplify problems involving repeated multiplication or division.

The objective is to rewrite the combined terms into a single logarithmic expression, making it easier to evaluate or understand the underlying growth patterns. Let's explore how the properties of logarithms assist in achieving this simplification.

Logarithm Properties

Logarithm properties are essential tools for simplifying and manipulating logarithmic expressions. These properties are derived from the rules of exponents. One fundamental property used in the original solution is known as the "Power Rule." This property allows you to move a coefficient (a number in front of a logarithm) as an exponent inside the logarithm. It states that \(a \log_b(x) = \log_b(x^a)\).

For instance, applying it to the expression \(4 \ln x\), we transform it into \(\ln x^4\). Similarly, \(7 \ln y\) becomes \(\ln y^7\), and \(3 \ln z\) turns into \(\ln z^3\). This step is crucial as it prepares the expression for further simplification using the product and quotient rules.

Understanding logarithm properties not only assists in solving specific mathematical problems but also enhances comprehension of the underlying structures of equations and relationships. These rules are often applicable in various subjects, including algebra, calculus, and real-world contexts like engineering and computer science.

Product and Quotient Rules

The product and quotient rules are pivotal in combining multiple logarithmic terms into a single expression. These rules hinge on the fundamental properties of exponents and how they interact with logarithms.

• Product Rule: This rule states that the sum of logarithms is equal to the logarithm of the product of their arguments: \(\log_b(x) + \log_b(y) = \log_b(x*y)\). This allows us to condense expressions such as \(\ln x^4 + \ln y^7\) into \(\ln (x^4*y^7)\).
• Quotient Rule: Similarly, the difference of two logarithms relates to the logarithm of their quotient: \(\log_b(x) - \log_b(y) = \log_b(\frac{x}{y})\). Using this rule, we can simplify \(\ln (x^4*y^7) - \ln z^3\) into a single logarithm \(\ln (\frac{x^4*y^7}{z^3})\).

Together, these rules are powerful in transforming complex combinations of logarithmic expressions into more manageable and comprehensible forms. Understanding the product and quotient rules can greatly simplify the process of handling exponential relationships and help reveal the structure and relationships within mathematical models.