Problem 47 - 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. \(\log 125\)

Step-by-Step Solution

Answer

The value of \(\log 125\) is approximately \(2.097\).

Step 1: Remember the Logarithm Power Rule

The power rule states that \(\log_b(a^n) = n \log_b(a)\). It allows us to shift from power operation to multiplication operation, letting us simplify complex logarithmic operations.

Step 2: Express 125 as a base to the power

Represent 125 as \(5^3\). Thus, \(\log 125\) becomes \(\log{(5^3)}\).

Step 3: Apply Logarithm Power Rule

By applying the power rule to \(\log{(5^3)}\) it simplifies to \(3 \log 5\).

Step 4: Substitute Given Logarithm Value

We replace \(\log 5\) with the given value \(0.6990\), so \(3 \log 5\) becomes \(3 \times 0.6990\).

Step 5: Multiply to Find the Result

Multiplying 3 by 0.6990, we get 2.097.

Key Concepts

Logarithm Power Rule

The logarithm power rule is a fundamental property when working with logarithms. It states that for any base \( b \), \( \log_b(a^n) = n \cdot \log_b(a) \). This rule transforms the challenge of dealing with powers into simple multiplication. For instance, if you need to evaluate \( \log(125) \) and express it as a power of a base, like \( 125 = 5^3 \), the power rule allows you to write this as \( 3 \cdot \log(5) \).

This powerful property becomes quite handy, especially when calculators are not allowed, and you need to evaluate logarithms with known log values. Using this rule maximizes your efficiency by breaking down complex exponentials into manageable parts that simply involve multiplication.

Logarithmic Expressions

Logarithmic expressions are equations that involve logarithms. They often require simplification for easier interpretation or calculation. In the context of the power rule, when provided with an expression like \( \log(125) \), it can initially seem daunting. However, knowing something about 125, such as its factorization to \( 5^3 \), you can apply the power rule to simplify it substantially.

This transformation is crucial: it turns a large number or complicated expression into something you can work with much more easily using known values. This concept recursively appears through all mathematics where restructuring an original complex problem to rely on simpler counterparts, helps find solutions quickly and accurately.

Simplifying Logarithms

To simplify logarithms, it's all about applying the right rules and using known values effectively. Given \( \log(5) \approx 0.6990 \), when we encounter \( \log(125) \), we can transform it from its intimidating form into a familiar friend. First, express 125 as the power \( 5^3 \). Then, utilize the power rule to break this down to \( 3 \cdot \log(5) \).

Now substitute the known value of \( \log(5) \), making it simply a matter of calculating \( 3 \times 0.6990 \) which results in 2.097. This process demonstrates the efficiency and elegance of simplifying logarithmic expressions by transforming them using properties we are already familiar with, concluding in results we can understand through basic arithmetic.