Problem 58 - Intermediate Algebra

Question

For Problems \(51-59\), you are given \(\log _{2} 5=2.3219\) and \(\log _{2} 7=2.8074\). Evaluate each expression using Properties 11.5-11.7. $$ \log _{2} 56 $$

Step-by-Step Solution

Answer

\( \log_{2} 56 = 2.8074 \)

Step 1: Express 56 as a Product

Notice that 56 can be expressed as a product of 5 and 11. Specifically, we have:\[ 56 = 5 \times 11 \]

Step 2: Apply the Product Property of Logarithms

The Product Property of Logarithms states that:\[ \log_b (MN) = \log_b M + \log_b N \]Applying this property, we can write:\[ \log_{2} 56 = \log_{2} 5 + \log_{2} 11 \]

Step 3: Use Given Values and Calculate Required Logs

We know from the problem statement that \( \log_{2} 5 = 2.3219 \) and \( \log_{2} 7 = 2.8074 \). We need to evaluate \( \log_{2} 11 \). Notice that:\[11 = \frac{7}{5}\]Therefore, \[\log_{2} 11 = \log_{2} \left(\frac{7}{5}\right)\]

Step 4: Apply the Quotient Property of Logarithms

The Quotient Property of Logarithms states that:\[ \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \]Therefore, we can write:\[ \log_{2} 11 = \log_{2} 7 - \log_{2} 5 \]

Step 5: Substitute Known Values

Substitute the given values into the equation: \[ \log_{2} 11 = 2.8074 - 2.3219 \]Calculate this difference:\[ \log_{2} 11 = 0.4855 \]

Step 6: Calculate \( \log_{2} 56 \) Using Values

Now, substitute back to find \( \log_{2} 56 \):\[ \log_{2} 56 = \log_{2} 5 + \log_{2} 11 \]Substitute the known values:\[ \log_{2} 56 = 2.3219 + 0.4855 \]Calculate this sum:\[ \log_{2} 56 = 2.8074 \]

Key Concepts

Product Property of Logarithms

The product property of logarithms is a useful technique that helps simplify complex expressions. When you have a logarithm of a product, like \( \log_b(MN) \), you can split it into a sum of two logarithms: \( \log_b M + \log_b N \). This property is based on the fundamental idea that multiplying numbers corresponds to adding their logarithms.
This property comes in handy when dealing with numbers that are products of smaller numbers. For example, to solve \( \log_{2} 56 \), we first express 56 as a product, such as \( 5 \times 11 \). Then we use the product property to transform the expression:

• \( \log_{2} 56 = \log_{2} (5 \times 11) = \log_{2} 5 + \log_{2} 11 \)

By breaking it down into smaller parts, evaluating logarithmic expressions becomes simpler, especially if you already know some component values.
Employing this property requires you to identify how to decompose your number effectively, making it pivotal for rewriting and solving complex logarithms.

Quotient Property of Logarithms

The quotient property of logarithms helps simplify expressions where a division appears inside a logarithm. According to this property, \( \log_b \left( \frac{M}{N} \right) \) can be expressed as \( \log_b M - \log_b N \). This relation exists because dividing corresponds to subtracting in the world of logarithms.
This is particularly helpful when you need to find a logarithm not directly given but can be represented as a quotient of other known logarithms. For example, to determine \( \log_{2} 11 \), which is not directly provided, we could express it using known values:

• Notice that \( 11 = \frac{7}{5} \), so \( \log_{2} 11 = \log_{2} \left( \frac{7}{5} \right) \)
• Using the quotient property, we rewrite it as \( \log_{2} 7 - \log_{2} 5 \)

With the values provided (\( \log_{2} 7 = 2.8074 \) and \( \log_{2} 5 = 2.3219 \)), we can calculate \( \log_{2} 11 \). This approach is essential when only partial data is available, transforming division within a logarithm to straightforward subtraction.

Evaluating Logarithmic Expressions

Evaluating logarithmic expressions involves combining various properties of logarithms to solve for unknown logarithmic values. This process often involves changing intricate logarithmic forms into simpler, computable parts.
Let’s consider how to evaluate an expression like \( \log_{2} 56 \) using the properties described above. Initially, express 56 as \( \log_{2} (5 \times 11) \), and employ the product property to write:

• \( \log_{2} 56 = \log_{2} 5 + \log_{2} 11 \)

Since \( \log_{2} 11 \) isn't directly provided, you can further use the quotient property by expressing 11 as \( \frac{7}{5} \) and finding:

• \( \log_{2} 11 = \log_{2} 7 - \log_{2} 5 \)

Substituting known values gives \( \log_{2} 11 = 2.8074 - 2.3219 = 0.4855 \).
By adding these evaluated parts back:

• \( \log_{2} 56 = 2.3219 + 0.4855 = 2.8074 \)

This method showcases how effectively using logarithmic properties allows you to solve expressions step-by-step, simplifying and gradually solving complex problems.