Problem 34 - Algebra 2 and Trigonometry

Question

Expand each expression using the properties of logarithms. \(\log _{4} \frac{x^{6}}{y^{5}}\)

Step-by-Step Solution

Answer

The expanded form is \( 6 \cdot \log_{4}(x) - 5 \cdot \log_{4}(y) \).

Step 1: Apply the Quotient Rule of Logarithms

The Quotient Rule for logarithms states that \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \). Apply this rule to the given expression: \( \log_{4} \frac{x^{6}}{y^{5}} = \log_{4}(x^{6}) - \log_{4}(y^{5}) \).

Step 2: Apply the Power Rule of Logarithms to Each Term

The Power Rule for logarithms states that \( \log_b (M^n) = n \cdot \log_b M \). Apply this to both terms in the expression: \( \log_{4}(x^{6}) = 6 \cdot \log_{4}(x) \) and \( \log_{4}(y^{5}) = 5 \cdot \log_{4}(y) \).

Step 3: Combine the Steps

Using the results from Steps 1 and 2, the expression becomes \( 6 \cdot \log_{4}(x) - 5 \cdot \log_{4}(y) \).

Key Concepts

Quotient Rule of Logarithms

When dealing with the division of two numbers within a logarithm, we can simplify the expression using the quotients' property of logarithms. This rule is a great tool for breaking down complex expressions into simpler parts. The Quotient Rule states:

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

It means if you have the logarithm of a division, you can rewrite it as a subtraction of two separate logarithms.
This can be especially handy when simplifying expressions or solving equations involving logarithms. For instance, if you have \( \log_4 \left( \frac{x^6}{y^5} \right) \), applying the Quotient Rule will split it into \( \log_4(x^6) - \log_4(y^5) \). This step sets the stage for further simplification using other rules like the Power Rule.

Power Rule of Logarithms

Once we have used the Quotient Rule, we can move forward with the Power Rule of logarithms. The Power Rule is useful when a logarithm contains an exponent. It allows us to deal with such powers more easily by moving them in front of the logarithm:

• \( \log_b(M^n) = n \cdot \log_b M \)

This rule really helps to simplify expressions where the arguments have been raised to a power. For instance, applying the Power Rule to \( \log_4(x^6) \) changes it to \( 6 \cdot \log_4(x) \). Similarly, \( \log_4(y^5) \) becomes \( 5 \cdot \log_4(y) \).
With everything laid out linearly, it’s now much easier to interpret and work with these logarithmic terms. It's important to remember that the exponent comes to the front as a multiplier, making calculations more straightforward.

Exponents in Logarithms

Exponents and logarithms often go hand in hand, especially when you're expanding or simplifying an expression. Understanding how exponents work within logarithms helps in transforming a complicated expression into a comprehensible form.
When an expression involves exponents, such as \( x^6 \) or \( y^5 \), incorporating the Power Rule allows us to bring down the exponent to the front of the logarithm. This pattern of breaking down expressions takes advantage of the properties of exponents:

• Not only does it make the expression easier to handle, but it also helps set the path for further mathematical operations.
• Once simplified, you can easily add, subtract, or equate logarithmic expressions, depending on what your problem asks for.

For example, in our expression \( \log_4 \left( \frac{x^6}{y^5} \right) \), elucidating the exponents using logarithms helps in neat presentation and easier computation, thereby demystifying what might otherwise seem complex.