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Chapter 9: Problem 8

Express as an equivalent expression that is a sum of logarithms. $$\log _{2}(16 \cdot 32)$$

Short Answer

Expert verified
The equivalent expression as a sum of logarithms is \(\log_2(16) + \log_2(32) = 4 + 5 = 9\).

Step by step solution

01

Apply the Product Rule of Logarithms

The product rule of logarithms states that \(\log_b(xy) = \log_b(x) + \log_b(y)\). Using this rule, separate the logarithm of the product into the sum of the logarithms: \(\log_2(16 \cdot 32) = \log_2(16) + \log_2(32)\).
02

Simplify Individual Logarithms Using Base Conversion

Now, simplify each of the logarithms. Recall that \(2^4 = 16\) and \(2^5 = 32\). Therefore, \(\log_2(16) = \log_2(2^4) = 4\) and \(\log_2(32) = \log_2(2^5) = 5\).
03

Sum Up the Simplified Logarithms

Add the simplified results obtained from Step 2: \(\log_2(16) + \log_2(32) = 4 + 5 = 9\).

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Key Concepts

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

Product Rule of Logarithms
The product rule of logarithms is a fundamental property used to break down logarithms of products into simpler terms. This rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. Formally, it is written as: \(\text{log}_b(xy) = \text{log}_b(x) + \text{log}_b(y)\). This property is very useful in simplifying complex logarithmic expressions. For instance, in our exercise, \(\text{log}_2(16 \times 32)\) can be expressed as \(\text{log}_2(16) + \text{log}_2(32)\). This transformation simplifies the original problem and makes it easier to solve. Remember to always identify the factors to which this rule can be applied.
Base Conversion
Understanding base conversion is crucial for simplifying logarithmic expressions. The base of a logarithm is the number that is raised to a power to produce a given number. For example, in the logarithmic expression \(\text{log}_2(16)\), the base is 2. To simplify this, you need to express 16 as a power of 2. Since \[16 = 2^4\], we get \[\text{log}_2(16) = \text{log}_2(2^4)\]. By using the property \[\text{log}_b(b^k) = k\], we find that \[\text{log}_2(2^4) = 4\]. Similarly, \(\text{log}_2(32)\) can be simplified to 5 because \[32 = 2^5\]. Thus, knowing how to convert numbers to the base of the logarithm is an essential step that aids in simplification.
Logarithmic Properties
Logarithmic properties include several rules that help simplify logarithmic expressions. These properties are based on the fundamental definitions and characteristics of logarithms. Here are a few key properties:
  • The product rule: \[\text{log}_b(xy) = \text{log}_b(x) + \text{log}_b(y)\]
  • The quotient rule: \[\text{log}_b(x/y) = \text{log}_b(x) - \text{log}_b(y)\]
  • The power rule: \[\text{log}_b(x^k) = k \cdot \text{log}_b(x)\]
  • The change of base formula: \[\text{log}_b(x) = \frac{\text{log}_c(x)}{\text{log}_c(b)}\]
In our exercise, we mainly used the product rule to transform \[\text{log}_2(16 \times 32) = \text{log}_2(16) + \text{log}_2(32)\]. Understanding and applying these properties greatly simplifies the process of working with logarithmic expressions. These rules help convert complex logarithmic equations into more manageable forms.

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