Problem 21 - Precalculus

Question

Write each expression as a sum or difference of logarithms. Example: \(\log \left(m^{2} n^{5}\right)=2 \log m+5 \log n\) $$\log _{b}\left(x^{3} y^{5}\right)$$

Step-by-Step Solution

Answer

\( \log_b(x^3 y^5) = 3 \log_b(x) + 5 \log_b(y) \).

Step 1: Identify the properties of logarithms

To break down the expression, we need to apply logarithm properties. We can use the power rule of logarithms, which states that \( \log_b(a^c) = c \cdot \log_b(a) \), and the product rule, which states that \( \log_b(mn) = \log_b(m) + \log_b(n) \).

Step 2: Apply the power rule

Apply the power rule to the expression \( \log_b(x^3 y^5) \). Separate the powers inside the logarithm: it becomes \( \log_b(x^3) + \log_b(y^5) \).

Step 3: Use the power rule individually

Now, apply the power rule to each logarithmic expression individually. For \( \log_b(x^3) \), use \( 3 \cdot \log_b(x) \). For \( \log_b(y^5) \), use \( 5 \cdot \log_b(y) \).

Step 4: Combine the results

Combine the results obtained from applying the power rule. The expression \( \log_b(x^3 y^5) \) is now written as \( 3 \log_b(x) + 5 \log_b(y) \).

Key Concepts

Power Rule of Logarithms

The power rule of logarithms is a straightforward yet powerful tool. It allows us to simplify logarithmic expressions when exponents are involved. Imagine you have a term like \( \log_b(a^c) \). This rule tells us we can "move" the exponent \( c \) in front of the logarithm as a multiplier. So, \( \log_b(a^c) \) becomes \( c \cdot \log_b(a) \).

This property is derived from the laws of exponents and helps in breaking down complex logarithmic terms into simpler, manageable parts. By using the power rule, you can efficiently transform exponential notation into a product, keeping calculations straightforward. This is especially useful for solving problems where variables appear with powers.

At its core, the power rule provides clarity and simplicity, making it easier for you to tackle expressions involving exponents. Use this as your first step in untangling any tricky logarithmic problem involving powers!

Product Rule of Logarithms

The product rule of logarithms is another fundamental property that helps simplify expressions. When you encounter a product inside a logarithm, like \( \log_b(mn) \), the product rule lets you split this into a sum of two separate logarithms: \( \log_b(m) + \log_b(n) \).

This rule is very handy because it lets us break down expressions into simpler components. By converting a product into a sum, you simplify the overall calculation. Logging a product is difficult for human understanding, but separating it into a sum helps align numbers in an arithmetic sequence, making other calculations much easier.

Think of this as slicing a big job into smaller tasks, making it more manageable. The product rule is essential in simplifying your work with logarithmic expressions, especially when combined with other logarithmic properties such as the power rule.

Logarithm Properties

Logarithm properties are crucial tools for transforming and simplifying expressions.

These properties include several key rules:

• The power rule, which transforms exponents within a log into a multiplier.
• The product rule, which turns a product inside a log into the sum of two logs.

These two rules are just a part of many. Others include the quotient rule, which is handy when dealing with fractions inside a logarithm by changing them into the difference between two logs.

Certainly, understanding and mastering these properties greatly ease the complexity involved in solving logarithmic equations. They provide clear paths to simplify and solve effectively. Knowing these properties not only helps in dealing with academic problems but also in real-world applications, as logarithms appear in many fields such as finance, physics, and computer science. Thus, these properties offer both theoretical and practical benefits.