Problem 46 - Algebra 2 and Trigonometry

Question

\(\operatorname{In} 36-47,\) if \(\log 3=x\) and \(\log 5=y,\) write each of the logs in terms of \(x\) and \(y\) $$ \log \frac{3}{5} $$

Step-by-Step Solution

Answer

\( \log \frac{3}{5} = x - y \)

Step 1: Understanding Logarithm Laws

To solve this problem, we must first recall the logarithm rules. One key rule is that the logarithm of a quotient can be expressed as the difference of the logarithms: \( \log \frac{a}{b} = \log a - \log b \). This will be useful for expressing \( \log \frac{3}{5} \) in terms of \( x \) and \( y \).

Step 2: Applying the Logarithm Quotient Rule

Apply the logarithm quotient rule to the given expression \( \log \frac{3}{5} \): \[ \log \frac{3}{5} = \log 3 - \log 5 \] We know that \( \log 3 = x \) and \( \log 5 = y \). Thus, we substitute these values into the equation.

Step 3: Substituting Known Values

Substitute \( \log 3 = x \) and \( \log 5 = y \) into the equation:\[ \log \frac{3}{5} = x - y \] Thus, \( \log \frac{3}{5} \) is expressed in terms of \( x \) and \( y \) as \( x - y \).

Key Concepts

Understanding Logarithm Laws

Logarithms are powerful tools in mathematics for dealing with exponential situations. They transform multiplication into addition, division into subtraction, and exponentiation into multiplication, which can simplify complex calculations. The three primary logarithm laws are:

• The Product Rule: \( \log(a \cdot b) = \log a + \log b \)
• The Quotient Rule: \( \log \frac{a}{b} = \log a - \log b \)
• The Power Rule: \( \log(a^b) = b \cdot \log a \)

These rules are derived from the properties of exponents since logarithms are essentially the inverse operations of exponents. For instance, the quotient rule is very useful when you have a division situation, as it allows you to break it down into a simpler subtraction problem in the world of logarithms.

Applying the Quotient Rule

The quotient rule is a handy shortcut for finding the logarithm of a ratio or fraction. If you have two positive numbers, \(a\) and \(b\), and you need to know \( \log \frac{a}{b} \), you can use the quotient rule to express this as \( \log a - \log b \).

This simplification is particularly useful because subtraction is often easier to handle than division. Let's consider the given expression \( \log \frac{3}{5} \). We can break it down using the quotient rule, transforming it into \( \log 3 - \log 5 \).

This breakdown allows you to further substitute known values for \( \log 3 \) and \( \log 5 \) if they are available, solidifying the solution to your logarithmic expression without direct computation of the log itself. This method is efficient, especially when specific logarithm values are given, as in this exercise.

Expressing Logarithmic Expressions in Terms of Other Variables

Using known values to find the expression of a logarithm in terms of other variables is another practical application of the logarithm laws. In this exercise, you're provided with \( \log 3 = x \) and \( \log 5 = y \).
By applying these given values, any logarithmic expression involving \(3\) and \(5\) can be easily expressed in terms of \(x\) and \(y\).

• For instance, using the expression \( \log \frac{3}{5} \), first apply the quotient rule to express it as \( \log 3 - \log 5 \).
• Then, substitute \( \log 3 = x \) and \( \log 5 = y \), resulting in \(x - y\).

In essence, the transformation of logarithmic expressions in terms of known variables not only simplifies computation but also prepares these expressions for further application in algebraic manipulations or problem-solving scenarios. By understanding how to use logarithm laws, students can effectively and efficiently solve complex logarithmic expressions.