Problem 67 - Algebra for College Students

Question

Given that \(\log _{8} 5=0.7740\) and \(\log _{8} 11=1.1531\), evaluate each expression using Properties \(10.5-10.7\) \(\log _{8}\left(\frac{25}{11}\right)\)

Step-by-Step Solution

Answer

The value of \( \log_{8}\left(\frac{25}{11}\right) \) is 0.3949.

Step 1: Set Up the Problem Using Logarithmic Properties

We are given the expression \( \log_{8} \left( \frac{25}{11} \right) \) and need to evaluate it using logarithmic properties. According to Property 10.6, \( \log_{b} \left( \frac{M}{N} \right) = \log_{b} M - \log_{b} N \). This allows us to express the problem as \( \log_{8} 25 - \log_{8} 11 \).

Step 2: Substitute Given Values and Simplify

To simplify our expression, we need \( \log_{8} 25 \), which can be rewritten using Property 10.5 as \( \log_{8} 5^2 = 2\log_{8} 5 \). Using the given value \( \log_{8} 5 = 0.7740 \), we can find \( 2 \times 0.7740 = 1.5480 \). Now substitute the known value for \( \log_{8} 11 \). Our expression becomes \( 1.5480 - 1.1531 \).

Step 3: Calculate Final Value

Perform the subtraction from the simplified expression: \( 1.5480 - 1.1531 = 0.3949 \). This is the value of \( \log_{8}\left(\frac{25}{11}\right) \).

Key Concepts

Logarithms

Logarithms are a fundamental concept in mathematics used to describe the power to which a number, called the base, must be raised to produce another number. For example, in the equation \( x = b^y\), \( y\) is the logarithm of \( x\) with base \( b\). Logarithms can simplify complex multiplication into simple addition, making them a valuable tool in various applications such as science, engineering, and finance.

• In logarithms, the base is a crucial part. It can be any positive real number except 1.
• The most common bases are 10 (common logarithm), \( e \) (natural logarithm), and 2 (binary logarithm).
• Logarithms turn multiplicative problems into additive ones, simplifying computations.

Understanding the properties of logarithms, such as the product, quotient, and power rules, helps us solve logarithmic expressions efficiently.

Base 8 Logarithm

A base 8 logarithm is a specific type of logarithm where the base is 8. This means we are looking at what exponent we need to raise 8 to get a certain number. For example, if \( \log_{8} x = y\), then \( 8^y = x\).

• Base 8 logarithms are not as common as base 10 or natural logarithms but are useful in certain mathematical and computational fields.
• They inherit all logarithm properties, allowing simplification of expressions using similar rules.

In the example given, you start with known logarithm values such as \( \log_8{5} = 0.7740\) and \( \log_8{11} = 1.1531\) to evaluate more complex expressions.

Logarithmic Expressions

Logarithmic expressions involve operations on logarithms, such as adding, subtracting, and applying logarithmic rules to simplify them. These expressions can look complicated, but using logarithmic properties makes them manageable.

• The problem \( \log_8\left(\frac{25}{11}\right)\) is a logarithmic expression.
• This expression can be separated into two simpler logarithmic terms using the quotient rule, effectively turning division into subtraction.

This approach allows us to rewrite and solve these expressions step by step. Using given values like those for \( \log_8{5}\) and \( \log_8{11}\), simplifies the process.

Logarithmic Identities

Logarithmic identities are equations that leverage the properties of logarithms to simplify complex expressions. Key identities include:

• Product Rule: \( \log_b(MN) = \log_b M + \log_b N \)
• Quotient Rule: \( \log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N \)
• Power Rule: \( \log_b(M^k) = k \cdot \log_b M \)

In the given exercise, these identities are used to break down the expression \( \log_8\left(\frac{25}{11}\right)\). The quotient rule is initially applied to separate the numerator and denominator, then the power rule helps further simplify \( \log_8{25} \) by expressing it as \( 2 \cdot \log_8{5} \). These steps transform complex expressions into more straightforward computations.