Problem 79 - Elementary and Intermediate Algebra

Question

Write logarithmic expression as one logarithm. \(-3 \log _{b} x-2 \log _{b} y+\frac{1}{2} \log _{b} z\)

Step-by-Step Solution

Answer

\(\log_b \left( \frac{z^{1/2}}{x^{3} y^2} \right)\) as one logarithm.

Step 1: Apply the Power Rule

Using the power rule of logarithms, which states that \( a \log_b M = \log_b M^a \), we can rewrite each term in the expression: - \(-3 \log_b x = \log_b (x^{-3})\),- \(-2 \log_b y = \log_b (y^{-2})\),- \(\frac{1}{2} \log_b z = \log_b (z^{1/2})\).

Step 2: Combine Logarithms Using Product Rule

Add and subtract the logarithms according to the given expression but using the product rule: \( \log_b (x^{-3}) + \log_b (z^{1/2}) - \log_b (y^{2})\).The product rule says \( \log_b M + \log_b N = \log_b (M \cdot N) \) and the subtraction rule \( \log_b M - \log_b N = \log_b (\frac{M}{N}) \).

Step 3: Simplify to a Single Logarithm

Now, combine all under a single logarithm: \[ \log_b \left( \frac{x^{-3} \cdot z^{1/2}}{y^2} \right). \] Combine and simplify:\[ \log_b \left( \frac{z^{1/2}}{x^{3} y^2} \right). \]

Key Concepts

Product Rule of Logarithms

The product rule of logarithms is a handy tool for combining multiple logarithmic expressions into one. This rule states that when you have the sum of two logs with the same base, you can combine them into a single log by multiplying the arguments. Formally, this can be expressed as:

• \( \log_b M + \log_b N = \log_b (M \cdot N) \)
• Similarly, if you have a difference, \( \log_b M - \log_b N = \log_b \left( \frac{M}{N} \right) \)

This is particularly useful when dealing with expressions that require simplification or consolidation into one logarithm. For example, if you need to combine \( \log_b (x^{-3}) + \log_b (z^{1/2}) - \log_b (y^{2}) \), you can use the product rule to write it as:\[ \log_b \left( \frac{x^{-3} \cdot z^{1/2}}{y^2} \right) \]By using the product rule, complex expressions can be simplified, making them easier to solve or further manipulate.

Power Rule of Logarithms

The power rule of logarithms allows you to bring a power in the logarithm argument out in front as a coefficient. This is expressed as:

• \( a \log_b M = \log_b M^a \)

This rule helps simplify logarithmic expressions where terms are multiplied by constant powers. For instance, when you encounter expressions like \(-3 \log_b x\), \(-2 \log_b y\), or \(\frac{1}{2} \log_b z\), you can rewrite them using the power rule:

• \(-3 \log_b x = \log_b (x^{-3})\)
• \(-2 \log_b y = \log_b (y^{-2})\)
• \(\frac{1}{2} \log_b z = \log_b (z^{1/2})\)

This conversion simplifies the process of combining logs later on. The power rule not only helps in simplifying complex expressions but also plays a critical role in transforming and manipulating logarithmic terms during calculation.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions is about making them as concise as possible by using the properties of logarithms, such as the product and power rules. Simplification involves several steps:

• **Identify and apply relevant rules**: Recognize the parts of the expression that can be manipulated using rules like the power rule to simplify powers into coefficients.
• **Combine terms using the product rule**: Once terms are simplified, use the product rule to combine them into a single logarithm.
• **Reorganize and simplify further**: Arranging terms and simplifying the argument of the logarithm to its simplest form, often consolidating expressions into a single fraction if necessary.

For the given expression, after applying the power rule and rearranging:\[ \log_b \left( \frac{z^{1/2}}{x^{3} y^2} \right) \]This expression is now a single, simplified logarithm. By breaking down the expression step-by-step and applying logarithmic rules, complex expressions become more manageable and easier to understand.