Problem 44 - Intermediate Algebra

Question

Write each logarithm as the sum and/or difference of logarithms of a single quantity. Then simplify, if possible. See Example 4. $$ \log \frac{9 t}{4} $$

Step-by-Step Solution

Answer

\( \log \frac{9t}{4} = \log 9 + \log t - \log 4 \).

Step 1: Identify the Expression

The expression given is \( \log \frac{9t}{4} \). This is a logarithm of a fraction.

Step 2: Apply the Quotient Rule of Logarithms

According to the quotient rule of logarithms, \( \log \frac{a}{b} = \log a - \log b \). For our expression: \[ \log \frac{9t}{4} = \log (9t) - \log 4 \].

Step 3: Apply the Product Rule of Logarithms

The product rule states that \( \log(ab) = \log a + \log b \). For \( \log (9t) \), we can use this to get: \[ \log (9t) = \log 9 + \log t \].

Step 4: Substitute Back into the Expression

Replace \( \log (9t) \) with its expanded form in the earlier expression: \[ \log \frac{9t}{4} = (\log 9 + \log t) - \log 4 \].

Step 5: Simplify the Expression

Reorganize the expression from Step 4: \[ \log \frac{9t}{4} = \log 9 + \log t - \log 4 \]. This is the expression written as the sum and difference of logarithms.

Key Concepts

Quotient Rule of Logarithms

When dealing with logarithms, especially those involving fractions, the quotient rule becomes extremely useful. This rule states that the logarithm of a division, such as \( \log \frac{a}{b} \), can be rewritten as the difference between two logarithms: \( \log a - \log b \).
This transformation is essential because it simplifies the comparison and manipulation of logarithms.

• To apply this rule, identify the numerator and the denominator of the fraction.
• Then, rewrite the logarithm of the entire fraction as a subtraction of the logarithm of the denominator from the logarithm of the numerator.

In our exercise, this is applied by recognizing \( \log \frac{9t}{4} \) as \( \log (9t) - \log 4 \). This is the first step in breaking down the logarithms into simpler components.

Product Rule of Logarithms

The product rule of logarithms helps when we have a product inside a logarithm, such as \( \log (ab) \). This rule says you can express it as the sum of two logarithms: \( \log a + \log b \).

• This technique is useful for breaking down complex expressions.
• Simply identify parts of the product, then decompose the logarithm using the rule.

In the context of our problem, this rule is applied to \( \log (9t) \) by considering it as \( \log 9 + \log t \).
This conversion is a step towards representing the logarithm in a simpler form, facilitating subsequent algebraic operations.

Simplifying Logarithms

After applying logarithm rules, the next significant step is simplifying the expression by combining and reorganizing terms.

• The key is to identify like terms and logical groupings to construct a concise final expression.
• In our example, this means substituting back all expanded terms from previous steps and ensuring clarity in the arrangement.

The ultimate goal is to express the logarithm as a sum and/or difference in its simplest form, as illustrated by \( \log 9 + \log t - \log 4 \).
This final configuration reveals the underlying structure of the initial logarithmic expression and is crucial for further mathematical processing or evaluation.