Problem 68 - 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. $$ \log _{8}\left(\frac{121}{25}\right) $$

Step-by-Step Solution

Answer

\( \log_8 \left( \frac{121}{25} \right) = 0.7582 \)

Step 1: Express as a Difference of Logarithms

Using the property \( \log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N \), express the given logarithmic expression as the difference of two logarithms: \[ \log_8 \left( \frac{121}{25} \right) = \log_8 121 - \log_8 25 \].

Step 2: Apply the Power Rule

Recognize that 121 and 25 are perfect squares: \( 121 = 11^2 \) and \( 25 = 5^2 \). Use the power rule for logarithms, \( \log_b M^n = n \cdot \log_b M \), to express \( \log_8 121 \) and \( \log_8 25 \) as:\[ \log_8 121 = \log_8 (11^2) = 2 \cdot \log_8 11 \] and \[ \log_8 25 = \log_8 (5^2) = 2 \cdot \log_8 5 \].

Step 3: Substitute Known Values

Insert the given logarithmic values for \( \log_8 5 \) and \( \log_8 11 \) into the equation:\[ \log_8 121 - \log_8 25 = 2 \cdot (1.1531) - 2 \cdot (0.7740) \].

Step 4: Simplify the Expression

Calculate the logarithmic expression by evaluating the multiplications and the subsequent subtraction:\[ 2 \cdot 1.1531 = 2.3062 \] and \[ 2 \cdot 0.7740 = 1.5480 \].Then, compute the difference: \[ 2.3062 - 1.5480 = 0.7582 \].

Key Concepts

Properties of Logarithms

Logarithms are a fundamental concept in mathematics that help us solve exponential equations by transforming them into linear forms. The properties of logarithms are rules that assist us in simplifying and evaluating logarithmic expressions.

These properties include:

• The **product rule**: This states that \( \log_b(M \times N) = \log_b M + \log_b N \). It indicates that the logarithm of a product can be rewritten as the sum of two separate logarithms.
• The **quotient rule**: Illustrated as \( \log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N \). This rule tells us that the logarithm of a quotient is the difference of the logarithms.
• The **power rule**: Expressed as \( \log_b M^n = n \cdot \log_b M \). This rule shows that a power within a logarithm can be brought out as a coefficient in front of the logarithm.

These properties are essential tools for simplifying complex logarithmic expressions, as they break down intricate operations into more manageable steps.
In our exercise, we utilize these properties sequentially to evaluate the given expression and find the solution.

Difference of Logarithms

The difference of logarithms, as described by the quotient rule, allows us to express the logarithm of a fraction as a difference between two logarithms. This can significantly simplify calculations, especially when known logarithm values are involved.

In our example, we're asked to evaluate \( \log_8\left(\frac{121}{25}\right) \). By applying the quotient rule, we can write this as:
\[ \log_8\left(\frac{121}{25}\right) = \log_8 121 - \log_8 25 \].
This transformation allows us to work separately with \( \log_8 121 \) and \( \log_8 25 \), simplifying each using other properties or provided logarithmic values.

The difference of logarithms is not just a mathematical trick—it is a potent tool for breaking down problems into finer, more manageable pieces. This approach makes working with logarithmic expressions much clearer and can be applied to a broad range of problems.

Power Rule for Logarithms

The power rule is pivotal when dealing with exponents within logarithmic expressions. It tells us that if the argument of a logarithm is raised to a power, this power can be moved in front of the logarithm as a multiplier.

Our problem involves the numbers 121 and 25, which can be rewritten as powers: \( 121 = 11^2 \) and \( 25 = 5^2 \). This allows us to apply the power rule:

• For \( \log_8 121 = \log_8 (11^2) \), we have \( 2 \cdot \log_8 11 \).
• Similarly, \( \log_8 25 = \log_8 (5^2) \) becomes \( 2 \cdot \log_8 5 \).

By using the known values \( \log_8 5 = 0.7740 \) and \( \log_8 11 = 1.1531 \), we substitute and calculate:
\[ 2 \cdot 1.1531 = 2.3062 \]
and
\[ 2 \cdot 0.7740 = 1.5480 \].
The power rule simplifies the expression, making it easier to handle mathematical operations and deductions.
Once simplified using the power rule, the final difference is easy to compute, giving us our answer of \( 0.7582 \).
This highlights how powerful logarithmic rules are in breaking down otherwise complex exponential equations into basic arithmetic.