Problem 54 - College Algebra

Question

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 5 \log _{6} x+6 \log _{b} y $$

Step-by-Step Solution

Answer

The final expression is \( \log _{6} x^{5} + \log _{b} y^{6} \)

Step 1: Identifying given expression

The given expression is \(5 \log _{6} x+6 \log _{b} y\)

Step 2: Use Power Rule of Logarithm

Power Rule of logarithm says that \(\log _{b} m^{n}=n \log _{b} m\). Using the power rule, we can detach the coefficients of the logs and place them as powers of the variables. Hence, the result would be \(\log _{6} x^{5} + \log _{b} y^{6}\)

Step 3: Condensing the logarithms

Remember, the sum of two logarithms with the same base can be written as a single logarithm using the product rule \(\log _{b} m + \log _{b} n = \log _{b} mn\). However, in our case we have different bases of logarithms. Therefore, it's not possible to condense this expression into a single logarithm because the bases of the two logarithms are different.

Step 4: Final step

The final expression, after applying the power rule, is \(\log _{6} x^{5} + \log _{b} y^{6}\)

Key Concepts

Power Rule of Logarithms

When dealing with logarithmic expressions, the power rule is an incredibly useful property. This rule states that for a logarithm of a number raised to a power, we can bring down the exponent and multiply it by the logarithm. In mathematical terms, this is expressed as \( \log_{b}(m^n) = n \cdot \log_{b}(m) \). In other words, when we see a constant multiplied by a logarithm, like \( 5 \log_{6} x \), we can rewrite this as \( \log_{6}(x^5) \).
The power rule helps simplify expressions as it allows us to transform multipliers into exponents, making it easier to work with different parts of the equation. This is exactly what was applied in Step 2 of the solution, where \( 5 \log_{6} x + 6 \log_{b} y \) was rewritten using the power rule, resulting in \( \log_{6}(x^5) + \log_{b}(y^6) \).
By recognizing the power rule, students can streamline calculations and tackle more complex logarithmic equations with confidence.

Condensing Logarithms

Condensing logarithms is another powerful technique used to simplify expressions. Generally, it involves combining multiple log terms into a single logarithm. The key formula here is \( \log_{b} m + \log_{b} n = \log_{b} (mn) \). This is known as the product rule of logarithms, allowing two logs with the same base to become a single log of the product of their arguments.
However, in our example \( 5 \log_{6} x + 6 \log_{b} y \), we encounter a unique situation. After applying the power rule, the expression becomes \( \log_{6}(x^5) + \log_{b}(y^6) \), but since the bases (\( 6 \) and \( b \)) are different, the sum cannot be condensed into one log. This highlights the importance of having the same base when trying to condense logarithms.
Understanding when and how we can condense log expressions is crucial for working with logarithms effectively. Students should always check the bases before attempting to condense.

Logarithmic Properties

Logarithmic properties provide fundamental tools for manipulating logarithms in various math problems. They include rules like the power rule, product rule, quotient rule, and change of base rule. These rules are essential for converting, simplifying, and computing both simple and complex logarithmic expressions.
Key properties include:

• Product Rule: \( \log_{b}(mn) = \log_{b}(m) + \log_{b}(n) \)
• Quotient Rule: \( \log_{b}\left(\frac{m}{n}\right) = \log_{b}(m) - \log_{b}(n) \)
• Power Rule: \( \log_{b}(m^n) = n \cdot \log_{b}(m) \)
• Change of Base Formula: \( \log_{b}(m) = \frac{\log_{k}(m)}{\log_{k}(b)} \), useful when bases need adjusting across different terms.

In practice, these properties allow for the transformation and solving of logarithmic problems. While we can't always combine terms due to different bases, recognizing these key properties remains beneficial for simplifying and solving problems. Mastering these fundamentals prepares students to handle a variety of logarithmic challenges efficiently.