Problem 63 - Intermediate Algebra

Question

For Problems \(60-68\), you are given \(\log _{8} 5=0.7740\) and \(\log _{8} 11=1.1531\). Evaluate each expression using Properties 11.5-11.7. $$ \log _{8} \sqrt{11} $$

Step-by-Step Solution

Answer

\( \log_8 \sqrt{11} = 0.57655 \).

Step 1: Identify the Expression

We need to evaluate the expression \( \log_8 \sqrt{11} \). This involves finding the logarithm of a square root of 11 with base 8.

Step 2: Apply the Property of Logarithms

Recall that the logarithm of a square root can be expressed using the property \( \log_b \sqrt{x} = \frac{1}{2} \log_b x \). For our problem, this translates to \( \log_8 \sqrt{11} = \frac{1}{2} \log_8 11 \).

Step 3: Use the Given Logarithmic Value

From the problem, we know \( \log_8 11 = 1.1531 \). Substitute this value into our expression to get \( \log_8 \sqrt{11} = \frac{1}{2} \times 1.1531 \).

Step 4: Perform the Calculation

Calculate \( \frac{1}{2} \times 1.1531 = 0.57655 \). Thus, \( \log_8 \sqrt{11} = 0.57655 \).

Key Concepts

Properties of Logarithms

Logarithms have several properties that make working with them easier. Understanding these properties can greatly simplify complex calculations:

• Product Property: The logarithm of a product is equal to the sum of the logarithms of the factors, i.e., \( \log_b (xy) = \log_b x + \log_b y \).
• Quotient Property: The logarithm of a quotient is equal to the difference of the logarithms, i.e., \( \log_b \left( \frac{x}{y} \right) = \log_b x - \log_b y \).
• Power Property: The logarithm of a power is equal to the exponent times the logarithm of the base, i.e., \( \log_b (x^n) = n\log_b x \).
• Root Property: The logarithm of a root can be calculated by dividing the logarithm of the base number by the index of the root. This property is helpful in our exercise, expressed as \( \log_b \sqrt{x} = \frac{1}{2} \log_b x \).

For example, if you have \( \log_8 \sqrt{11} \), use the root property, allowing you to express it as half of \( \log_8 11 \). This simplification is key to solving the problem effectively.

Base Conversion

Changing the base of a logarithm is sometimes necessary when dealing with calculations or comparisons that require a different logarithmic base. There is a specific formula for converting logarithms from one base to another, which is known as the Change of Base Formula:

• Change of Base Formula: \( \log_b a = \frac{\log_k a}{\log_k b} \), where \( k \) can be any positive number different from 1.

Using this formula, you can convert a logarithm to a different base, such as base 10 or base \( e \), which are more common on calculators. For example, if you need to convert \( \log_8 11 \) using base 10, you would compute \( \log_{10} 11 / \log_{10} 8 \). This formula ensures you accurately transform your logarithmic expression without altering the value.

Square Roots

At its core, a square root is a value that, when multiplied by itself, gives the original number. For instance, the square root of 4 is 2 because \( 2 \times 2 = 4 \). Square roots are often represented with the radical symbol \( \sqrt{} \) or as an exponent of 1/2, for example, \( x^{1/2} \).
When it comes to applying properties of logarithms to square roots, the root property of logarithms becomes incredibly useful. This property allows you to simplify logarithmic expressions involving square roots. For example, the expression \( \log_8 \sqrt{11} \) translates to \( \frac{1}{2} \cdot \log_8 11 \) because taking the square root is analogous to raising a number to the power of 1/2.
Understanding how to break down square roots through logarithms helps streamline complex equations, particularly in scenarios where you need precise computation accuracy, such as solving mathematical exercises or real-world problems.