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Chapter 4: Problem 124

$$\text { Expand: } \log _{b}(x \sqrt[3]{y})$$

Short Answer

Expert verified
\(\log_b(x) + \frac{1}{3}\log_b(y)\

Step by step solution

01

Apply the Logarithmic Product Rule

Identify that the term inside the logarithm is a product of 'x' and the cube root of 'y'. Using the product rule for logarithms, this can be written as: \(\log_b(x) + \log_b(\sqrt[3]{y})\). The product rule states that the logarithm of a product is equal to the sum of the logarithm of each factor.
02

Apply the Logarithmic Power Rule

Observe that \(\log_b(\sqrt[3]{y})\) can be written as \(\log_b (y^{1/3})\), which is an instance of the logarithmic power rule (logarithm of a power of a number equals the exponent times the logarithm of the number). Utilizing the power rule, \(\log_b (y^{1/3})\) can be written as \(\frac{1}{3}\log_b (y)\).
03

Combine the Parts

By combining the results from steps 1 and 2, the expanded form of the original expression can be written as: \(\log_b(x) + \frac{1}{3}\log_b(y)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Logarithmic Product Rule
When dealing with logarithms, the product rule is a powerful tool. This rule helps us break down the logarithm of a product into the sum of the logarithms of individual factors. In mathematical terms, if you have
  • \( \log_b(MN) \)
this can be expanded to
  • \( \log_b(M) + \log_b(N) \).
Let's break it down. If you see a multiplication inside a logarithm, you can "split" them, making it easier to simplify or expand. In our original problem, we had \( \log_b(x \sqrt[3]{y}) \), where \( \sqrt[3]{y} \) stands for the cube root of \( y \). Using the product rule, we rewrite it as:
  • \( \log_b(x) + \log_b(\sqrt[3]{y}) \).
This first step helps in organizing terms for further simplification.
Exploring the Logarithmic Power Rule
The power rule allows us to take the exponent of a term in a logarithm and move it out front as a multiplier. It's especially useful for breaking down complex logarithmic expressions. If you encounter something like
  • \( \log_b(N^k) \),
you can rewrite it using the rule as
  • \( k \cdot \log_b(N) \).
In our exercise, we had \( \log_b(\sqrt[3]{y}) \), which can be expressed as \( \log_b(y^{1/3}) \). By applying the power rule:
  • \( \log_b(y^{1/3}) = \frac{1}{3} \times \log_b(y) \).
This step helps us transform powers into more manageable coefficients, simplifying the logarithmic expression further.
Expanding Logarithmic Expressions
Expanding logarithms is about taking a complex expression and breaking it down into simpler, more manageable parts. By using both the product and power rules, we can deal with intricate terms inside logarithms. Let's look again at our expression:
  • \( \log_b(x \sqrt[3]{y}) \).
First, apply the product rule, splitting it into:
  • \( \log_b(x) + \log_b(\sqrt[3]{y}) \).
Then, use the power rule to change \( \log_b(\sqrt[3]{y}) \) into \( \frac{1}{3} \cdot \log_b(y) \). Here's the expanded form:
  • \( \log_b(x) + \frac{1}{3} \log_b(y) \).
By expanding, we're able to simplify and solve logarithmic problems more efficiently, making them less intimidating and easier to work with.

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