Problem 251 - Precalculus

Question

For the following exercises, expand each logarititim as much as possible. Rewite each expression as a sum, difference, or product of logs. $$ \log _{b}(7 x \cdot 2 y) $$

Step-by-Step Solution

Answer

\(\log_{b}7 + \log_{b}x + \log_{b}2 + \log_{b}y\)

Step 1: Identify the Expression Structure

The given expression is the logarithm of a product, \(\log_{b}(7x \cdot 2y)\). Notice that it involves multiplying two terms: \(7x\) and \(2y\).

Step 2: Apply the Logarithm of a Product Rule

The logarithm of a product can be expressed as the sum of the logarithms: \(\log_{b}(a \cdot c) = \log_{b}a + \log_{b}c\). Apply this rule: \(\log_{b}(7x \cdot 2y) = \log_{b}(7x) + \log_{b}(2y)\).

Step 3: Expand Each Logarithm with Multiplication

Each component like \(\log_{b}(7x)\) and \(\log_{b}(2y)\) contains a product. Use the product rule again: \(\log_{b}(7x) = \log_{b}7 + \log_{b}x\) and \(\log_{b}(2y) = \log_{b}2 + \log_{b}y\).

Step 4: Combine All Parts

Put all expanded logs together: \(\log_{b}(7x \cdot 2y) = \log_{b}7 + \log_{b}x + \log_{b}2 + \log_{b}y\).

Key Concepts

Logarithm of a Product

When dealing with logarithms, understanding how to handle products is essential. The rule for logarithms of a product simplifies expressions significantly. In essence, when you take the logarithm of a product, you can separate this into the sum of logarithms of the individual factors. This is given by the formula: \[ \log_{b}(a \cdot c) = \log_{b}a + \log_{b}c \] For example, in the expression \( \log_{b}(7x \cdot 2y) \), you can apply this rule to break it down. By applying the logarithm of a product rule, you separate it into two parts, giving you \( \log_{b}(7x) + \log_{b}(2y) \). This rule is particularly useful for simplifying complex logarithmic expressions.

• It turns a single, potentially complex logarithm into multiple simpler logarithms.
• It helps in calculations where each factor can be simplified individually.

Understanding this rule is foundational for mastering logarithms and will assist in dealing with more complicated expressions later on.

Logarithmic Identities

Logarithmic identities are mathematical formulas that provide rules and shortcuts for manipulating or simplifying logarithm expressions. Becoming familiar with these is incredibly helpful when expanding logs or rewriting them. For instance, the identity for expansion of a product into a sum, already discussed, is just one of several useful identities. Some other valuable identities include:

• **Logarithm of a Quotient**: \( \log_{b}(\frac{a}{c}) = \log_{b}a - \log_{b}c \)
• **Logarithm of a Power**: \( \log_{b}(a^c) = c \cdot \log_{b}a \)

By memorizing these identities, you can transform complex problems into simpler forms and thus, solve them more easily. These along with the product rule are central to breaking down logarithmic expressions.

Properties of Logarithms

The properties of logarithms simplify complicated expressions and make it easier to work with equations involving logarithms. They are crucial tools whether you are solving equations or need to find an easier expression for calculations. Some fundamental properties to remember include:

• **Product Property**: \( \log_{b}(a \cdot c) = \log_{b}a + \log_{b}c \)
• **Quotient Property**: \( \log_{b}(\frac{a}{c}) = \log_{b}a - \log_{b}c \)
• **Power Property**: \( \log_{b}(a^c) = c \cdot \log_{b}a \)
• **Change of Base Formula**: \( \log_{b}a = \frac{\log_{k}a}{\log_{k}b} \), where \( k \) is any positive number

These properties also explain why logarithms can sometimes appear in condensed forms and how they can be expanded. They help demystify why certain transformations are possible when dealing with logarithms, and why the work shown in the step by step solution is valid. Understanding these properties will improve your overall comprehension and capability to manage logarithmic expressions and equations.