Problem 40 - Algebra 2 and Trigonometry

Question

Write each expression in terms of \(A\) and \(B\) if \(\log _{2} x=A\) and \(\log _{2} y=B\). \(\log _{2}(x \div y)\)

Step-by-Step Solution

Answer

The expression \( \log_{2}(x \div y) \) is \( A - B \).

Step 1: Understand the expression

We are given the expression \( \log_{2}(x \div y) \). Our job is to rewrite this expression using \( A \) and \( B \), where \( A = \log_{2} x \) and \( B = \log_{2} y \).

Step 2: Applying the logarithm quotient rule

The logarithm quotient rule states that \( \log_{b}(\frac{M}{N}) = \log_{b}M - \log_{b}N \). Apply this rule to \( \log_{2}(x \div y) \): \[ \log_{2}(x \div y) = \log_{2}(x) - \log_{2}(y) \]

Step 3: Substitute given expressions

We know that \( \log_{2} x = A \) and \( \log_{2} y = B \). Substitute these into the expression: \[ \log_{2}(x) - \log_{2}(y) = A - B \]

Step 4: Conclusion

Thus, the expression \( \log_{2}(x \div y) \) in terms of \( A \) and \( B \) is \( A - B \).

Key Concepts

Logarithm Quotient Rule

The logarithm quotient rule is an essential property of logarithms that makes dealing with logarithmic expressions involving division straightforward and efficient. This rule states that for any positive numbers \(M\) and \(N\), and a logarithm with a base \(b\), the following equation holds:

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

This means you can express the logarithm of a quotient as the difference of two simpler logs.
Whenever you encounter a logarithmic expression involving divided terms, the quotient rule offers a simple method to break it down.
In practice, the quotient rule allows you to transform complex expressions into more manageable pieces, making calculations easier. Using our exercise example, \(\log_{2}(x \div y)\), applying this rule tells us:

• \(\log_{2}(x) - \log_{2}(y)\)

Now, with terms defined as \(A = \log_{2}x\) and \(B = \log_{2}y\), we can substitute these into our rewritten expression for further clarity.

Change of Base Formula

The change of base formula is another powerful tool in the study of logarithms, especially when you need to evaluate logs with bases that aren't easily calculated. This formula allows conversion of a logarithm from one base to any other base:

• To convert \(\log_{b}M\) to base \(k\): \(\log_{b}M = \frac{\log_{k}M}{\log_{k}b}\)

Using this formula can simplify difficult logarithmic evaluations, especially when calculators or tables only support certain bases, such as base 10 or the natural logarithm base \(e\).
The formula relies on the equality between different logarithmic expressions and can be useful in various mathematical contexts, although it wasn't directly applied in our original exercise example.
Understanding this formula creates a strong foundation for learning how different logarithmic rules connect and complement one another.

Logarithmic Expressions

Logarithmic expressions might seem complicated at first, but they become much easier once you get familiar with their properties and how to manipulate them using various rules.
These expressions involve a combination of numbers, variables, and logarithms, and can often be simplified using key logarithmic rules such as the quotient rule mentioned earlier.
When faced with something like \(\log_{2}(x \div y)\), it's crucial to understand each piece of the expression:

• \(\log_{b}\) specifies the logarithm with a base \(b\)
• The division \(x \div y\) indicates a quotient in the argument
• Our roles for \(A\) and \(B\) boil down these logs to simpler terms for computation

Learning to work with logarithmic expressions not only builds algebra skills but also opens doors to solving more complex equations and functions in advanced mathematics.
As you master these expressions, you'll gain confidence in identifying when and how to apply different rules.