Problem 63 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.)$$\log _{b} \frac{x^{4} \sqrt{y}}{z^{5}}$$.

Step-by-Step Solution

Answer

The expanded form of the expression is \(4\log_{b} x + \frac{1}{2} log_{b} y - 5 log_{b} z\).

Step 1: Apply Quotient Rule

The first action to take is to apply the quotient rule of logarithms. The given expression has the form: \(\log_{b} \frac{x^{4} \sqrt{y}}{z^{5}}\).This can be broken down into: \(\log_{b} x^{4}\sqrt{y} - \log_{b} z^{5}\).

Step 2: Apply Product Rule

Apply the product rule of logarithms to the first part of the expression. The rule allows splitting of logarithmic expressions in multiplication form into an addition of separate logarithms. The expression: \(\log_{b} x^{4}\sqrt{y}\) can be broken down into: \(\log_{b} x^{4} + \log_{b} \sqrt{y}\). So, the overall expression after step 2 becomes: \(\log_{b} x^{4} + \log_{b}\sqrt{y} - \log_{b} z^{5}\).

Step 3: Apply Power Rule

Apply the power rule of logarithms to each part of the expression. The power rule allows extraction of the exponents and makes them coefficients of the separate logs. Thus, \(\log_{b} x^{4}\), \(\log_{b}\sqrt{y}\), \(\log_{b} z^{5}\) become \(4\log_{b} x\), \(\frac{1}{2} \log_{b} y\) and \(5 \log_{b} z\) respectively. Joining back, we get: \(4\log_{b} x + \frac{1}{2} \log_{b} y - 5 \log_{b} z\).

Key Concepts

Quotient Rule of Logarithms

The quotient rule of logarithms is a fundamental principle that assists in simplifying complex logarithmic expressions. This rule is particularly useful when dealing with the logarithm of a fraction. In plain terms, it states that the logarithm of a quotient is the difference between the logarithms of the numerator and the denominator.

Mathematically, it can be represented as:

• \[ \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \]

For instance, in the problem provided, we use the quotient rule to break down \( \log_b \frac{x^4 \sqrt{y}}{z^5} \) into two separate terms: \( \log_b (x^4 \sqrt{y}) - \log_b (z^5) \).

By employing this rule, we effectively simplify a complex logarithmic fraction into more manageable parts, making further calculations easier.

Product Rule of Logarithms

The product rule of logarithms is another vital tool for simplifying logarithmic sentences that involve multiplication. This rule states that the logarithm of a product is equal to the sum of the logarithms of the factors.

The mathematical expression of this rule is:

• \[ \log_b (MN) = \log_b M + \log_b N \]

In our exercise, after applying the quotient rule, we faced the expression: \( \log_b x^4 \sqrt{y} \). To simplify it further, we apply the product rule, which allows us to express it as \( \log_b x^4 + \log_b \sqrt{y} \). This simplification transforms our expression into a more straightforward form that prepares it for subsequent steps like the application of the power rule.

Using the product rule breaks down complex products into simpler, additive components, greatly aiding in clarity and ease of calculation.

Power Rule of Logarithms

The power rule of logarithms offers an efficient way to deal with logarithms of expressions where variables are raised to exponents. This rule allows us to take the exponent and multiply it as a coefficient in front of the logarithm, simplifying expressions considerably.

The power rule is expressed as follows:

• \[ \log_b (M^n) = n \cdot \log_b M \]

In the context of our problem, after separating terms with the quotient and product rules, we'll apply this power rule to \( \log_b x^4 \), \( \log_b \sqrt{y} \), and \( \log_b z^5 \). By converting the square root into a power, we rewrite \( \sqrt{y} \) as \( y^{1/2} \).

This leads to:

• \( 4 \cdot \log_b x \) for \( x^4 \)
• \( \frac{1}{2} \cdot \log_b y \) for \( \sqrt{y} \)
• \( 5 \cdot \log_b z \) for \( z^5 \)

Applying the power rule simplifies the process of handling larger or more intricate powers, turning them into manageable steps used along with other logarithmic rules.