Problem 48 - Algebra A Combined Function

Question

Write each expression as a sum or difference of logarithms. Assume that variables represent positive numbers. $$ \log _{2} y^{3} z $$

Step-by-Step Solution

Answer

The expression is written as \(3 \cdot \log_2(y) + \log_2(z)\).

Step 1: Apply the Power Rule for Logarithms

The power rule for logarithms states that \( \log_b(x^n) = n \cdot \log_b(x) \). Apply this rule to \( y^3 \) to rewrite the expression as \[ \log_2(y^3) + \log_2(z) = 3 \cdot \log_2(y) + \log_2(z) \].

Step 2: Combine the Expressions

The expression \( 3 \cdot \log_2(y) + \log_2(z) \) is already the sum of logarithms. Check to ensure all rules have been applied, and there are no further simplifications applicable.

Key Concepts

Power Rule for Logarithms

The power rule for logarithms is a very handy tool when you're dealing with exponents inside a logarithm. This rule states that for any positive number and real logarithmic base, the logarithm of a number raised to a power can be simplified. If you have an expression like \( \log_b(x^n) \), you can apply this rule to rewrite it as \( n \cdot \log_b(x) \). In simple terms, you "bring down" the exponent as a multiplier in front of the logarithm.
Applying this to our example, \( \log_2(y^3) \), we can rewrite it as \( 3 \cdot \log_2(y) \). This simplification makes complex expressions easier to handle. The key takeaway is that the power in an exponential expression can be moved in front of the logarithm to simplify calculations.

Sum and Difference of Logarithms

Combining logarithms using addition or subtraction is based on key properties of logarithms. Essentially, when you multiply two numbers inside a log, you can rewrite it as a sum of two separate logs. On the other hand, if you're dividing, you can write it as the difference of the logs.
For example, if you have \( \log_b(mn) \), this becomes \( \log_b(m) + \log_b(n) \), representing the sum. This is useful when dealing with expressions that contain products inside a logarithm.
For division, \( \log_b\left(\frac{m}{n}\right) \) results in \( \log_b(m) - \log_b(n) \), where you subtract the logs instead. In our solved problem, the expression \( \log_2(y^3) + \log_2(z) \) is already simplified as a sum because each component is separate. Understanding these operations allows easier manipulation of logarithmic expressions.

Logarithm Properties

Logarithm properties form the foundation for manipulating and understanding logarithmic expressions. Key properties include the product rule, quotient rule, power rule, and change of base formula.

• **Product Rule**: \( \log_b(mn) = \log_b(m) + \log_b(n) \)
• **Quotient Rule**: \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \)
• **Power Rule**: \( \log_b(x^n) = n \cdot \log_b(x) \)
• **Change of Base Formula**: \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \) for any positive base \( c \)

These properties are essential for simplifying expressions and solving logarithmic equations. They provide ways to break down complex problems into manageable parts. For example, in our solved problem, we utilized the power rule to handle the exponent inside the log. By understanding each of these properties, students can approach logarithmic questions with clarity and confidence.