Problem 271 - Calculus Volume 1

Question

Use properties of logarithms to write the expressions as a sum, difference, and/or product of logarithms. \(\log _{3} \frac{9 a^{3}}{b}\)

Step-by-Step Solution

Answer

\( \log_{3}(9) + 3\log_{3}(a) - \log_{3}(b) \).

Step 1: Apply the Quotient Rule

Use the quotient rule of logarithms, which states that \( \log_b \left( \frac{M}{N} \right) = \log_b(M) - \log_b(N) \). So, apply the rule to the expression \( \log_{3} \frac{9a^3}{b} \). This results in \( \log_{3}(9a^3) - \log_{3}(b) \).

Step 2: Apply the Product Rule

Next, apply the product rule to the term \( \log_{3}(9a^3) \). The product rule states \( \log_b(MN) = \log_b(M) + \log_b(N) \). In this case, \( 9a^3 \) is a product, so \( \log_{3}(9a^3) = \log_{3}(9) + \log_{3}(a^3) \).

Step 3: Apply the Power Rule

Use the power rule for logarithms on the term \( \log_{3}(a^3) \). The power rule states \( \log_b(M^n) = n \cdot \log_b(M) \). Thus, \( \log_{3}(a^3) = 3 \cdot \log_{3}(a) \).

Step 4: Combine the Results

Combine all the transformed terms. Substitute \( \log_{3}(9) + \log_{3}(a^3) - \log_{3}(b) \) with the result from applying the power rule: \( \log_{3}(9) + 3 \cdot \log_{3}(a) - \log_{3}(b) \). This is the final expression as a sum and difference of logarithms.

Key Concepts

Quotient Rule

The quotient rule of logarithms is a handy tool when you are faced with a logarithmic expression of a fraction. In simple terms, it helps you break down complex fractions into subtractive parts.
When you have a term like \( \log_b \left( \frac{M}{N} \right) \), you can transform it using the quotient rule into \( \log_b(M) - \log_b(N) \). This tells you that the logarithm of a division is the difference of the logarithms.

• "M" is the numerator, and "N" is the denominator.
• It converts a division problem inside a logarithm into a subtraction problem of two simpler logs.

In our exercise, you see this applied in the first step: \( \log_{3} \frac{9a^3}{b} = \log_{3}(9a^3) - \log_{3}(b) \).
Now you have two separate logs, making it easier to tackle the expression with further logarithmic properties.

Product Rule

When dealing with the product of two numbers inside a logarithm, the product rule becomes your best friend. This rule allows you to split an intricate multiplication within logs into additive segments.
It states that for any base \( b \), the formula \( \log_b(MN) = \log_b(M) + \log_b(N) \) holds. This means the logarithm of a product is the sum of the logarithms of its factors.

• "M" and "N" are the factors you are multiplying together inside the log.
• This rule helps simplify multiplication problems into addition, which is generally easier to manage.

In our original exercise, apply this to \( \log_{3}(9a^3) \), resulting in \( \log_{3}(9) + \log_{3}(a^3) \).
This step brings you closer to breaking the expression down into even simpler parts using other properties like the power rule.

Power Rule

The power rule of logarithms helps you when you encounter an exponent within a logarithmic expression. It assists in moving the exponent out front, turning a potentially complicated multiplication into a straightforward one.
According to this rule, \( \log_b(M^n) \) becomes \( n \cdot \log_b(M) \). This is powerful because it changes a power inside a log into a coefficient outside.

• "n" represents the exponent.
• This rule simplifies powers into multiplication, making calculations more manageable.

In our step-by-step solution, notice this application on \( \log_{3}(a^3) \), which converts it to \( 3 \cdot \log_{3}(a) \).
By applying the power rule, the entire problem becomes simpler and fits neatly with other rules to give a clearer and manageable final expression: \( \log_{3}(9) + 3 \cdot \log_{3}(a) - \log_{3}(b) \).