Problem 90 - Intermediate Algebra

Question

Assume that \(\log 4 \approx 0.6021, \log 7 \approx 0.8451,\) and \(\log 9 \approx 0.9542 .\) Use these values to evaluate each logarithm. See Example 8 . $$ \log _{b} 36 $$

Step-by-Step Solution

Answer

\(\log_{b} 36 = \frac{1.5563}{\log b}\); further evaluation requires \(\log b\).

Step 1: Understanding the Base 36 Logarithm

The given problem is to find the logarithm of 36 with an unspecified base \(b\). We can write \(\log_b 36\) as \(\frac{\log 36}{\log b}\) using the change of base formula, which states \(\log_b A = \frac{\log A}{\log b}\). To utilize this, we need expressions for \(\log 36\) and \(\log b\) in terms of the provided logs.

Step 2: Express 36 as a Product of Known Factors

To find \(\log 36\), we express 36 as a product of known bases: \(36 = 4 \times 9\). Using the property of logarithms that \(\log A \times B = \log A + \log B\), we can compute \(\log 36\) as \(\log 4 + \log 9\).

Step 3: Calculate \(\log 36\)

Given \(\log 4 \approx 0.6021\) and \(\log 9 \approx 0.9542\), we can now compute \(\log 36\) using the expression \(\log 36 = \log 4 + \log 9\). Thus, \(\log 36 \approx 0.6021 + 0.9542 = 1.5563\).

Step 4: Evaluate the Expression if Possible

Since no specific \(\log b\) value has been provided and we have expressed \(\log 36\), the evaluation generally remains as \(\log_b 36 = \frac{1.5563}{\log b}\). Without knowledge of \(\log b\), we cannot compute a specific numerical value. The problem may imply you demonstrate forming but not necessarily solving for a definitive numeric answer.

Key Concepts

Change of Base Formula

The change of base formula is a valuable tool when dealing with logarithms that don't have a readily apparent base. This formula allows us to express a logarithm in terms of logs with different bases, typically base 10 or base e (natural logarithms). Let's break this down.

The formula is written as:

• \( \log_b A = \frac{\log A}{\log b} \)

This means that the logarithm of A with base b can be rewritten as the ratio of the logarithm of A to the logarithm of b, both using a common logarithm like base 10. This method is beneficial for performing calculations on calculators that typically only have keys for common logarithm (log base 10) and natural logarithm (log base e).

Using this approach, you can compute logs with awkward bases by converting them into a more manageable form. Remember, the key idea is to have both logs - \(\log A\) and \(\log b\) - computed in the same base for consistency.

Properties of Logarithms

The properties of logarithms extend your ability to manipulate and simplify logarithmic expressions effectively. One of the core properties you should always keep in mind is the product property, which states:

• \( \log(A \cdot B) = \log A + \log B \)

This property allows us to break down more complex logarithmic terms into manageable pieces, as seen in finding \(\log 36\), expressed as \(4 \times 9\). By using the property, it becomes clear that:

• \( \log 36 = \log 4 + \log 9 \)

Additional properties worth knowing include the quotient property \(\log \left( \frac{A}{B} \right) = \log A - \log B\) and the power property \(\log(A^n) = n \cdot \log A\). Each helps to transform and simplify logarithmic expressions in various contexts.

Understanding these properties not only aids in calculations but also deepens your grasp of the relationships within logarithmic equations.

Logarithmic Expressions

Logarithmic expressions may initially seem complex, but they become quite manageable with the right approach and understanding of foundational concepts. In the example with \(\log 36\), you learned how to decompose the problem using known values.

• Given \(\log 4 \approx 0.6021\) and \(\log 9 \approx 0.9542\), \(\log 36 = \log 4 + \log 9\).
• Adding these, \(\log 36 \approx 1.5563\).

Breaking complex problems into simpler parts using known logarithmic expressions is key. Also, maintaining familiarity with core properties like the associative, commutative, and distributive laws in multiplication and addition provides you with the flexibility needed to tackle different mathematical challenges.

With practice, you'll feel more comfortable transforming logarithmic expressions, knowing that each expression holds a wealth of smaller parts that can be interchanged and added together to reveal new insights.