Problem 64 - Algebra for College Students

Question

Given that \(\log _{8} 5=0.7740\) and \(\log _{8} 11=1.1531\), evaluate each expression using Properties \(10.5-10.7\) \(\log _{8}(5)^{2 / 3}\)

Step-by-Step Solution

Answer

0.5160

Step 1: Apply the Power Property

The power property of logarithms states that \( \log_b (m^n) = n \cdot \log_b (m) \). Here, we apply this property with \( m = 5 \) and \( n = \frac{2}{3} \). Thus, \( \log_{8}(5)^{2/3} = \frac{2}{3} \cdot \log_{8}(5) \).

Step 2: Substitute Known Values

Use the given value \( \log_{8}5 = 0.7740 \) and substitute it into the expression from Step 1. This gives us \( \log_{8}(5)^{2/3} = \frac{2}{3} \cdot 0.7740 \).

Step 3: Perform the Multiplication

Calculate \( \frac{2}{3} \cdot 0.7740 \). This involves multiplying 0.7740 by 2 and then dividing by 3. \( 0.7740 \times 2 = 1.5480 \), and \( 1.5480 \div 3 = 0.5160 \).

Step 4: Final Evaluation

The value of \( \log_{8}(5)^{2/3} \) is 0.5160. Thus, using the properties of logarithms, the expression evaluates to 0.5160.

Key Concepts

Power Property of Logarithms

The power property of logarithms is a useful tool for simplifying logarithmic expressions. This property states that for any positive real number \( m \) and a real number \( n \), \( \log_b(m^n) = n \cdot \log_b(m) \). It essentially allows us to "take the exponent down" and multiply it with the logarithm of the base number.
To apply this property, start by identifying the base \( b \), the exponent \( n \), and the number \( m \) being raised to the power. For instance, in the expression \( \log_{8}(5)^{2/3} \), \( m = 5 \) and \( n = \frac{2}{3} \).

• The exponent \( \frac{2}{3} \) comes down in front of the logarithm, simplifying the expression to \( \frac{2}{3} \cdot \log_{8}(5) \).

This property is particularly helpful when the log of a base is already known and you need to evaluate an expression where that base is raised to a power.

Base Conversion

In logarithms, base conversion refers to changing the base of a logarithm to one that is more convenient for calculation. This is often necessary when expressing a logarithm in terms of known values of logs with different bases.
Although base conversion is not directly used in evaluating \( \log_{8}(5)^{2/3} \), understanding it can help when working with problems involving multiple bases. The change of base formula is \( \log_b(m) = \frac{\log_k(m)}{\log_k(b)} \), where \( k \) is any positive real number different from 1.

• Commonly, \( k \) is chosen as 10 or \( e \) (Euler's number) because these are the bases of the common and natural logarithms, respectively.
• This conversion can make complex calculations more manageable by converting to bases for which logarithmic values are readily available or easier to compute.

When dealing with exercises that involve changing bases, being familiar with this conversion formula simplifies the process.

Logarithm Evaluation

Evaluating a logarithmic expression involves using known properties and sometimes substituting known values. Based on the given values \( \log_{8}5 = 0.7740 \) and \( \log_{8}11 = 1.1531 \), the evaluation becomes streamlined.
In our exercise, once we simplify using the power property, we substitute the known value directly.

• First, calculate the value, knowing that \( \log_{8}(5) = 0.7740 \)
• Use the simplified expression \( \log_{8}(5)^{2/3} = \frac{2}{3} \times 0.7740 \).
• Perform the multiplication step-by-step: First, multiply 0.7740 by 2 to get 1.5480, then divide by 3 to reach a final result of 0.5160.

The process of evaluating logs often involves similar strategic substitutions and arithmetic operations, making use of given log values or tables.