Problem 57 - Introductory and Intermediate Algebra for College Students 4th

Question

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions. $$3 \ln x+5 \ln y-6 \ln z$$

Step-by-Step Solution

Answer

The condensed logarithmic expression is \( \ln(x^3y^5 / z^6) \)

Step 1: Use the Power Rule

Start by applying the power rule of logarithms, which states that \(\log_b (a^n) = n \log_b a\). Each term of the expression \(3 \ln x+5 \ln y-6 \ln z\) can be rewritten with the logarithm taken to a power. This gives: \(\ln(x)^3 + \ln(y)^5 - \ln(z)^6\).

Step 2: Apply the Product and Quotient Rules

The next step involves using the Product Rule and the Quotient Rule. The Product Rule states \(\log_b (mn) = \log_b m + \log_b n\), and the Quotient Rule states \(\log_b (m/n) = \log_b m - \log_b n\). This allows us to rewrite the expression as a single logarithm: \(\ln(x^3y^5 / z^6)\).

Step 3: Final Answer

The expression is now fully condensed and the solution is \(\ln(x^3y^5 / z^6)\). It cannot be evaluated any further without given values for \(x, y, z\).

Key Concepts

Properties of Logarithms

Logarithms come with unique properties that make them extremely useful in simplifying expressions and solving equations. Understanding these properties helps in condensing terms into a single, simplified logarithmic expression. Some of the fundamental properties to remember include:

• **Product Rule:** Allows you to combine logs by multiplying the values inside the logarithm: \(\log_b (mn) = \log_b m + \log_b n\).
• **Quotient Rule:** Helps in condensing division inside a log: \(\log_b (m/n) = \log_b m - \log_b n\).
• **Power Rule:** Lets you express the exponent as a coefficient: \(\log_b (a^n) = n \log_b a\).

In the context of simplifying expressions, these properties enable us to transform terms efficiently. The goal is often to express an equation as a single logarithm, streamlining calculation and analysis. Always keep these rules in your toolkit when approaching logarithmic problems.

Power Rule of Logarithms

The Power Rule of logarithms is a tool that helps handle exponents within a logarithm. It allows you to take an exponent from inside a logarithm and transform it into a coefficient, simplifying the expression. The rule is expressed as:\[ \log_b (a^n) = n \log_b a \]
This means if you have a term like \(3 \ln x\), it can be rewritten using the power rule as \(\ln(x^3)\). This transformation is facilitated by recognizing the link between exponents and multiplication in logs.
Applying this rule helps break down complex expressions. For example, in the exercise given, terms such as \(3 \ln x\) become more manageable as \(\ln(x^3)\). This condensed representation simplifies further manipulation of logarithmic equations, making it a helpful technique in algebraic simplification.

Product Rule of Logarithms

The Product Rule of logarithms provides a method to combine multiple logarithmic expressions into a single log by multiplying the arguments of the logarithms. This rule is particularly useful when dealing with the sum of separate logs. It states:
\[ \log_b (mn) = \log_b m + \log_b n \]
This rule becomes pivotal when simplifying complex logarithmic expressions to a single form. For example, if you have \(\ln(x^3) + \ln(y^5)\), applying the product rule combines these into \(\ln(x^3 y^5)\).
By using the product rule, you can condense multiple logarithms into one, mitigating the complexity of the expression and simplifying further operations or evaluations. This not only makes expressions more compact but also simplifies subsequent mathematical processes.

Quotient Rule of Logarithms

The Quotient Rule of logarithms assists in merging terms involving division into a single logarithmic expression by indicating how to subtract logarithms. It is described by:
\[ \log_b (m/n) = \log_b m - \log_b n \]
In the context of the given problem, this rule combines outputs of the power rule when those outputs are subtracted. For instance, if you hold \(\ln(x^3 y^5) - \ln(z^6)\), applying the quotient rule transforms it to \(\ln(x^3 y^5 / z^6)\).
This rule is crucial for simplifying expressions with division. By using the quotient rule, we can convert expressions into concise, single-logarithm forms. This efficiency aids in solving equations and understanding relationships within logarithmic contexts. Understanding and applying the quotient rule is key to mastering logarithmic transformations.