Problem 520 - Precalculus

Question

Condense the expression \(3 \log _{7} v+6 \log _{7} w-\frac{\log _{7} u}{3}\) a single logarithm.

Step-by-Step Solution

Answer

\( \log_{7} \left(\frac{v^3 w^6}{u^{1/3}}\right) \)

Step 1: Use Logarithm Multiplication Rule

Begin by recognizing that the expressions are sums and differences of logarithms, which can be combined using logarithmic properties. The expression is \[ 3 \log_{7} v + 6 \log_{7} w - \frac{\log_{7} u}{3}. \]Use the property \( a \log_{b} x = \log_{b} x^a \) to rewrite each term:\[ \log_{7} v^3 + \log_{7} w^6 - \log_{7} u^{1/3}. \]

Step 2: Combine Using Addition Rule

Combine the logs involved in addition to a single log using the property \( \log_{b} x + \log_{b} y = \log_{b} (xy) \). Apply this to the first two terms:\[ \log_{7} (v^3 w^6) - \log_{7} u^{1/3}. \]

Step 3: Use Subtraction Rule to Condense

Now, apply the logarithm subtraction rule \( \log_{b} x - \log_{b} y = \log_{b} \left(\frac{x}{y}\right) \) to condense the expression into one log:\[ \log_{7} \left(\frac{v^3 w^6}{u^{1/3}}\right). \]

Key Concepts

Logarithm Rules

Logarithm rules are a set of fundamental principles that help simplify and manipulate logarithmic expressions. These rules are essential for working with expressions that involve logarithms. The primary rules include:

• Multiplication Rule: This tells us how to handle the sum of logarithms.
• Subtraction Rule: This helps us with the difference of two logarithms.
• Power Rule: This involves using an exponent to change the form of a logarithmic expression.

These rules are powerful because they allow us to "condense" or combine multiple logarithmic terms into a single, more manageable expression. Mastery of these rules is crucial for solving logarithmic problems efficiently.

Multiplication Rule

The multiplication rule for logarithms allows you to combine the logarithms of two products into a single logarithmic expression. This is often presented as:

• When you have two or more logarithms with the same base that are added together, they can be condensed into a single logarithm of the product of the arguments.

Mathematically, the rule is expressed as \( \log_b x + \log_b y = \log_b (xy) \).
This rule simplifies the process of working with complex expressions.
In our example, this rule allowed the terms \( \log_{7} v^3 + \log_{7} w^6 \) to be simplified to \( \log_{7} (v^3 w^6) \) by treating the multiplication inside the log as addition of the logs themselves.

Subtraction Rule

The subtraction rule for logarithms is similarly useful and applies when you are dealing with the difference of two logarithmic expressions.
This rule shows that:

• The subtraction of two logarithms with the same base is equal to a single logarithm of the division of the arguments.

In mathematical terms, this is written as \( \log_b x - \log_b y = \log_b \left(\frac{x}{y}\right) \).
In the provided problem, the subtraction rule was used to simplify the expression \( \log_{7} (v^3 w^6) - \log_{7} u^{1/3} \) into a single logarithm expression \( \log_{7} \left(\frac{v^3 w^6}{u^{1/3}}\right) \).
This transforms a complex, multi-logarithm problem into a much easier one.

Condensing Expressions

Condensing expressions in logarithmic terms means reducing multiple logarithmic terms into a single expression. This involves applying the above rules—the multiplication, subtraction, and power rules.

• The goal is to simplify the expression so it is easier to work with and understand.

In the given exercise, we went from several logs expressing powers and divisions, to just one neat expression: \( \log_{7} \left(\frac{v^3 w^6}{u^{1/3}}\right) \).
This process involves taking each step—using the multiplication and subtraction rules—and applying them step-by-step. Each step makes the expression tidier, easier to read, and more efficient for calculation or further manipulation.