Problem 50 - Algebra 2

Question

Assume that \(\log 4 \approx 0.6021, \log 5 \approx 0.6990,\) and \(\log 6 \approx 0.7782 .\) Use the properties of logarithms to evaluate each expression. Do not use your calculator. \(\log \frac{1}{4}\)

Step-by-Step Solution

Answer

-0.6021

Step 1: Applying Logarithmic Property

The first step is to apply the logarithmic property that states \(\log \frac{1}{a} = - \log a\). Using this rule, the expression \(\log \frac{1}{4}\) can be rewritten as \(- \log 4\).

Step 2: Substitute the Given Logarithm

Next, substitute the given value of \(\log 4\) into the equation. This was given in the problem as \(\log 4 \approx 0.6021\). By substituting this into the transformed equation from Step 1, the expression becomes \(-0.6021\).

Step 3: Final Conclusion

Given that there are no further algebraic manipulations to perform, the final value is therefore \(-0.6021\).

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is crucial in simplifying and evaluating logarithmic expressions without using a calculator. One key property is the power rule, which states that \( \log_b(a^c) = c \cdot \log_b a \). Another is the product rule: \( \log_b(mn) = \log_b m + \log_b n \), and the quotient rule: \( \log_b\left(\frac{m}{n}\right) = \log_b m - \log_b n \).

In the exercise, the property \( \log \frac{1}{a} = - \log a \) is used. This is derived from the quotient rule and reflects that the logarithm of a reciprocal can be transformed into a negative logarithm. This property is especially useful when given a logarithmic value for the non-reciprocal form, as it allows for straightforward calculations without a calculator.

These properties are foundational because they transform complex expressions into simpler ones, often converting multiplication into addition or division into subtraction, which are much easier operations to perform mentally.

Logarithmic Transformations

Logarithmic transformations help simplify complex logarithmic expressions by converting them into a more manageable form. These transformations are applications of the properties of logarithms to rewrite expressions.

In the given exercise, the expression \( \log \frac{1}{4} \) is transformed using the property \( \log \frac{1}{a} = - \log a \). This converts the original expression into \( - \log 4 \), a much simpler form to evaluate, given that \( \log 4 \) is known from the problem statement.

Such transformations are powerful because they allow us to break down complex expressions or equations into simpler components. Practice in these transformations enhances algebra skills and builds a deep understanding of logarithmic behavior. Remembering and utilizing these transformations make it easier to solve a wide range of math problems involving logarithms.

Evaluating Logarithmic Expressions

Evaluating logarithmic expressions involves determining the exact or approximate value of the expression using known values and logarithmic properties.

In problems like the one given, we start by identifying if any known logarithmic values can be used directly or indirectly. The key is recognizing transformations required to simplify the expression, such as the property \( \log \frac{1}{4} = - \log 4 \).

Once transformed, we substitute the known values: here, \( \log 4 \approx 0.6021 \), to easily compute \( -0.6021 \) for \( \log \frac{1}{4} \). This process eliminates the need for a calculator by leveraging properties and known values.

By practicing the evaluation of logarithmic expressions, students can sharpen their problem-solving skills. This involves identifying properties, making transformations, and accurately substituting known values, skills that are essential for success in mathematics.