Problem 49 - Elementary and Intermediate Algebra

Question

Write logarithm as the sum and/or difference of logarithms of a single quantity. Then simplify, if possible. \(\log _{8} \frac{1}{8 m}\)

Step-by-Step Solution

Answer

\(-1 - \log_8(m)\)

Step 1: Apply the Quotient Rule for Logarithms

The quotient rule for logarithms states that \(\log_b \left( \frac{M}{N} \right) = \log_b(M) - \log_b(N)\). Apply this to the expression \(\log_8 \left( \frac{1}{8m} \right)\). Thus, it becomes: \[\log_8(1) - \log_8(8m)\]

Step 2: Simplify \(\log_8(1)\)

Recalling that any logarithm of 1 in any base is 0, so \(\log_8(1) = 0\). The expression is now simplified to: \[0 - \log_8(8m)\]

Step 3: Apply the Product Rule for Logarithms

According to the product rule, \(\log_b(MN) = \log_b(M) + \log_b(N)\). Apply this to \(\log_8(8m)\) and the expression becomes: \[- (\log_8(8) + \log_8(m))\]

Step 4: Evaluate \(\log_8(8)\)

The logarithm \(\log_b(b)\) is always 1. Thus, \(\log_8(8) = 1\). Substitute this into the expression: \[- (1 + \log_8(m)) = -1 - \log_8(m)\]

Step 5: Write the Final Expression

The simplified expression for \(\log_8 \frac{1}{8m}\) can be written as a sum and/or difference of logarithms: \[-1 - \log_8(m)\]

Key Concepts

Quotient Rule for Logarithms

When working with logarithmic expressions, the quotient rule is a handy tool for breaking down complex logarithms into simpler parts. The quotient rule for logarithms states:

• \( \log_b \left( \frac{M}{N} \right) = \log_b(M) - \log_b(N) \)

This rule is extremely useful when you encounter a logarithm of a division. It allows you to express the logarithm of a quotient as the difference between two separate logarithms. This is especially helpful when simplifying or transforming expressions in algebra.

For example, if you have \( \log_8 \left( \frac{1}{8m} \right) \), applying the quotient rule splits it into two terms: \( \log_8(1) \) and \(-\log_8(8m) \). You can see how this process transforms a single complicated logarithm into more manageable pieces.

Understanding this rule simplifies the evaluation and manipulation of logarithms, allowing for more streamlined problem-solving processes.

Product Rule for Logarithms

Another important rule in simplifying logarithmic expressions is the product rule. This rule helps when you encounter products inside the logarithm. The product rule states:

• \( \log_b(MN) = \log_b(M) + \log_b(N) \)

Using this rule, you can separate a single logarithm of a product into the sum of two logarithms. This is a fantastic way to break down products into simpler logarithms, making complex expressions easier to deal with.

In the expression \(- \log_8(8m)\), applying the product rule further simplifies it to \(- (\log_8(8) + \log_8(m))\). This step reveals the individual parts of the product, turning what was initially a complex expression into pieces that are easier to evaluate and understand.

Such simplifications not only aid in calculations but also deepen the comprehension of how logarithms function within equations.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions often requires applying both the quotient and product rules effectively. In addition to these, it's crucial to remember key logarithmic identities, like \(\log_b(b) = 1\) and \(\log_b(1) = 0\).

In our example, once you apply these rules and identities, you reach the expression \(-1 - \log_8(m)\).

• First, apply the quotient rule to handle the division inside the log.
• Next, apply the identity \(\log_8(1) = 0\), reducing the expression further.
• Then, use the product rule to separate the product inside the log.
• Finally, use the identity \(\log_8(8) = 1\) to replace and simplify the terms.

Through systematic application of these rules and identities, the expression becomes much simpler and more insightful. Each simplification step not only makes the expression easier to work with but also provides a clearer view of the relationships within the logarithmic and algebraic structures involved.