Problem 56 - Algebra for College Students

Question

Given that \(\log _{2} 5=2.3219\) and \(\log _{2} 7=2.8074\), evaluate each expression by using Properties \(10.5-10.7\) \(\log _{2} \sqrt[3]{5}\)

Step-by-Step Solution

Answer

\(\log_{2} \sqrt[3]{5} \approx 0.774\)

Step 1: Understand the Problem

We need to evaluate \(\log_{2} \sqrt[3]{5}\) using given logarithmic values and properties of logarithms.

Step 2: Use the Power Rule

We apply Property 10.6, the power rule for logarithms, which states that \(\log_b (a^c) = c \cdot \log_b a\). Here, \(\sqrt[3]{5}\) can be written as \(5^{1/3}\).

Step 3: Apply the Power Rule

Using the power rule, we have:\[\log_{2} \left(\sqrt[3]{5}\right) = \log_{2} \left(5^{1/3}\right) = \frac{1}{3} \cdot \log_{2} 5\].

Step 4: Substitute the Known Value

Substitute \(\log_{2} 5 = 2.3219\) into the expression:\[\log_{2} \left(\sqrt[3]{5}\right) = \frac{1}{3} \cdot 2.3219\].

Step 5: Calculate the Expression

Now, calculate the value:\[\frac{1}{3} \cdot 2.3219 = 0.77397\].

Key Concepts

Logarithmic Properties

Logarithmic properties are essential tools that simplify the process of solving logarithmic equations. They offer systematic methods to manipulate and evaluate logarithms. These properties are quite similar in importance to arithmetic properties like adding and multiplying numbers. Some key logarithmic properties are:

• Product Rule: This states \[\log_b(MN) = \log_b M + \log_b N\]\
  It helps you split the logarithm of a product into the sum of separate logarithms.
• Quotient Rule: Another important property is \[ \log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N\] \
  It allows you to break down the logarithm of a division.
• Power Rule: This states that \[ \log_b(M^c) = c \cdot \log_b M\] \, letting you take exponents out of the logarithm as a factor. This property was used in our exercise to deal with the cube root by expressing it as an exponent.

Understanding these properties is key. They simplify the calculations and shed light on the relationships between numbers and their logarithms.

Power Rule

The power rule for logarithms is a handy property that helps us deal with logarithms, especially those involving roots and exponents. This property states: \(\log_b (a^c) = c \cdot \log_b a\). It means that the exponent, denoted by \(c\), can be taken as a multiplier outside the logarithm. This rule is significant when dealing with expressions where the base or the arguments are raised to a power.In our exercise, the expression \(\log_{2} \sqrt[3]{5}\) is simplified using the power rule.

• The cube root \(\sqrt[3]{5}\) is rewritten as an exponent: \(5^{1/3}\).
• Applying the power rule changes it to \(\frac{1}{3} \cdot \log_{2} 5\).

Thus, effectively breaking down a complex logarithm into a simpler arithmetic operation. This step is crucial as it directly utilizes the given logarithmic value of base 2 for 5 and simplifies the entire evaluation process.

Change of Base Formula

The change of base formula is an invaluable technique, especially when the logarithm's base is not convenient for computation. This formula allows conversion of logarithms between different bases, thereby facilitating easier calculation with calculators that often support logarithms to base 10 or base \(e\) (natural logarithm). The formula is:\[ \log_b a = \frac{\log_k a}{\log_k b} \]Where \(k\) is a new base you want to use. This formula is powerful because:

• It enables calculations with any base by translating it into a common base, like 10 or \(e\).
• This conversion is extremely useful when dealing with unconventional bases or distributing computations.

Although not explicitly used in the original exercise, understanding this concept is essential for problems where exotic bases are involved or when you only have access to base 10 or natural logarithms.