Problem 8 Express as an equivalent express... [FREE SOLUTION]

Chapter 12: Problem 8

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

Short Answer

Expert verified

\[ \log_{2}(16 \cdot 32) = \log_{2}(16) + \log_{2}(32) = 4 + 5 = 9 \]

Step by step solution

Apply the product rule of logarithms

Use the product rule for logarithms, which states that $\log_b (xy) = \log_b(x) + \log_b(y)$. For the given expression $ \log_{2}(16 \cdot 32) $, apply this rule.

Identify and separate the factors

Identify the factors inside the logarithm. In this case, the factors are 16 and 32. Therefore, $\log_{2}(16 \cdot 32) $ can be written as $\log_{2}(16) + \log_{2}(32)$.

Simplify each logarithm term

Evaluate each logarithm term individually. $\log_{2}(16)$ is the logarithm of 16 to the base 2. Since \16 = 2^4\, $\log_{2}(16) = 4$. Similarly, $\log_{2}(32)$ is the logarithm of 32 to the base 2. Since \32 = 2^5\, $\log_{2}(32) = 5$.

Combine the results

Add the simplified logarithm terms together. The expression becomes $\log_{2}(16) + \log_{2}(32) = 4 + 5$, which simplifies to 9.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

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 identity that allows you to break down a logarithm of a product into a sum of logarithms. It states that $\log_b (xy) = \log_b(x) + \log_b(y)$. This rule is essential when dealing with complex logarithmic expressions as it makes them easier to handle by splitting them into simpler parts.
For example, in the problem $\log_2(16 \cdot 32)$, we can apply the product rule:

Identify the factors inside the logarithm: 16 and 32.
Express the product as a sum of logarithms: $\log_2(16) + \log_2(32)$.

This simplifies the problem significantly and makes it easier to evaluate each logarithm term separately.

logarithmic simplification

Logarithmic simplification involves reducing the complexity of logarithmic expressions by using logarithmic identities and properties. Once we've applied the product rule, the next step is to simplify each individual logarithmic term.
For example, consider $\log_2(16)$ and $\log_2(32)$:

Recognize that 16 can be written as $2^4$. Hence, $\log_2(16) = 4$ because $\log_2(2^4) = 4$.
Similarly, 32 can be expressed as $2^5$, so $\log_2(32) = 5$ because $\log_2(2^5) = 5$.

By expressing the factors in terms of the base (2 in this case), we can simplify the logarithmic terms effortlessly.

logarithmic identities

Understanding and utilizing logarithmic identities is crucial for simplifying logarithmic expressions. Some key logarithmic identities that are often used include:

The product rule: $\log_b (xy) = \log_b(x) + \log_b(y)$.
The power rule: $\log_b (x^y) = y\log_b(x)$.
The change of base formula: $\log_b (x) = \frac{\log_k (x)}{\log_k (b)}$, which allows you to change the base of the logarithm.

These identities can be applied in various steps to make simplifying logarithmic expressions manageable.
In our exercise, after applying the product rule, we simplify each term using the power rule (recognizing expressions like $\log_2(2^4) = 4$ and $\log_2(2^5) = 5$). Finally, we combine the simplified terms to get the result: $4 + 5 = 9$. Knowing these identities helps solve logarithmic problems more effectively and makes the overall process straightforward.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

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

Short Answer

Step by step solution

Apply the product rule of logarithms

Identify and separate the factors

Simplify each logarithm term

Combine the results

Key Concepts

product rule of logarithms

logarithmic simplification

logarithmic identities

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Applied Mathematics

Statistics

Logic and Functions

Geometry

Pure Maths

Study anywhere. Anytime. Across all devices.