Problem 73 - A Graphical Approach to Precalculus with Limits

Question

Use the properties of logarithms to rewrite each logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{k} \frac{p q^{2}}{m}$$

Step-by-Step Solution

Answer

\( \log_k p + 2 \cdot \log_k q - \log_k m \)

Step 1: Apply the Quotient Rule

Use the quotient rule of logarithms which states that \( \log_b \frac{x}{y} = \log_b x - \log_b y \). Apply this to split the original logarithm: \( \log_k \frac{pq^2}{m} = \log_k (pq^2) - \log_k m \).

Step 2: Apply the Product Rule

Next, apply the product rule of logarithms to \( \log_k (pq^2) \), using \( \log_b (xy) = \log_b x + \log_b y \). We have: \( \log_k (pq^2) = \log_k p + \log_k q^2 \).

Step 3: Apply the Power Rule

Now, use the power rule on \( \log_k q^2 \) by using \( \log_b x^n = n \cdot \log_b x \). Thus, \( \log_k q^2 = 2 \cdot \log_k q \).

Step 4: Combine All Parts

Put all the pieces together. Start with the result from Step 1: \( \log_k \frac{pq^2}{m} = \log_k (pq^2) - \log_k m \). Substitute using steps 2 and 3: \( \log_k p + 2 \cdot \log_k q - \log_k m \).

Key Concepts

Quotient Rule of Logarithms

The quotient rule for logarithms helps us break down more complex logarithmic expressions into simpler components. This rule states:

• \( \log_b \frac{x}{y} = \log_b x - \log_b y \)

When you have to deal with the logarithm of a quotient, this rule becomes extremely useful. You can express the logarithm of a division as the difference of two separate logarithms. For example, given the expression \( \log_k \frac{pq^2}{m} \), we apply the quotient rule and split it into:

• \( \log_k (pq^2) - \log_k m \)

This step makes it easier to handle the remaining parts of the expression. Remember, when applying the quotient rule, make sure both the numerator and the denominator are positive real numbers. The quotient rule helps in simplifying expressions, making calculations more manageable.

Product Rule of Logarithms

The product rule of logarithms simplifies the logarithim of a product of factors by expressing it as the sum of individual logarithms. It’s defined as follows:

• \( \log_b (xy) = \log_b x + \log_b y \)

This rule is particularly handy when you deal with products inside logarithmic expressions. For instance, in the expression \( \log_k (pq^2) \), applying the product rule gives:

• \( \log_k (pq^2) = \log_k p + \log_k q^2 \)

Using the product rule helps separate elements within a multiplicative relationship into individual parts, making further simplification easier. Applying this rule requires you to break down the product into smaller parts, helping you manage complex expressions more efficiently. Before using the product rule, check that all variables involved are positive real numbers.

Power Rule of Logarithms

The power rule of logarithms allows you to tackle expressions where a variable is raised to a power. According to this rule, an expression with an exponent can be rewritten as a multiplication:

• \( \log_b x^n = n \cdot \log_b x \)

This rule is instrumental when dealing with terms that have powers. For example, in \( \log_k q^2 \), the power rule lets you simplify the expression to:

• \( 2 \cdot \log_k q \)

The power rule is useful when you need to express the logarithmic term as a multiple of another term, facilitating easier calculations and solutions. Always confirm that the base is positive and not equal to one, and ensure that the exponent is valid within the mathematical context.