Problem 75 - Precalculus with Limits

Question

Condensing a Logarithmic Expression In Exercises \(67-82,\) condense the expression to the logarithm of a single quantity. $$\log x-2 \log y+3 \log z$$

Step-by-Step Solution

Answer

The logarithmic expression \( \log x - 2 \log y + 3 \log z \) can be condensed to a single quantity as \( \log(\frac{x z^3}{y^2}) \)

Step 1: Rewrite the expression

To condense the logarithmic expression to a single quantity, we'll use the properties of logarithms. The expression is currently written as \( \log x - 2 \log y + 3 \log z \)

Step 2: Applying power rule

The power rule states that \( \log b^n = n \log b \). Applying the power rule to the expression, we get \( \log x - \log y^2 + \log z^3 \)

Step 3: Applying sum and difference rule

The sum rule states that \( \log a + \log b = \log(ab) \). And the difference rule states that \( \log a - \log b = \log(\frac{a}{b}) \). Now by applying these rules, we get the expression as \( \log x - \log y^2 + \log z^3 = \log(\frac{x z^3}{y^2}) \)

Step 4: Final expression

Thus, the logarithmic expression \( \log x - 2 \log y + 3 \log z \) can be condensed to a single quantity as \( \log(\frac{x z^3}{y^2}) \)

Key Concepts

Logarithmic Expressions

Logarithmic expressions are mathematical phrases that involve logarithms. A logarithm is the inverse operation to exponentiation, just like subtraction is the inverse of addition. For example, consider the expression \( \log x \), where \( x \) is called the argument. In simple terms, a logarithm answers the question: "To what power must a certain base be raised, to produce a given number?". Logarithmic expressions often come in more complex forms like \( \log x - 2 \log y + 3 \log z \), which is a combination of several logarithms. To work with these, it's important to simplify or condense them, which typically makes analysis easier.

Properties of Logarithms

Properties of logarithms are essential to manipulating and simplifying logarithmic expressions. Knowing these properties allows you to condense or expand expressions efficiently:

• Product Rule: \( \log a + \log b = \log(ab) \) - Use this when combining logarithms that are added.
• Quotient Rule: \( \log a - \log b = \log(\frac{a}{b}) \) - Use this to combine logarithms that are subtracted.
• Power Rule: \( n \log b = \log b^n \) - Convert the multiplication of a logarithm by a number into an exponentiation of the argument.

These properties are crucial for transforming expressions like \( \log x - 2 \log y + 3 \log z \) into a form that is easier to understand and work with.

Logarithmic Rules

Logarithmic rules are fundamental tools in algebra, particularly when dealing with exponential growth and decay, among other applications. The rules enable the transformation and simplification of logarithmic expressions. When you see an expression such as \( \log x - 2 \log y + 3 \log z \), the rules tell us how to condense it:

• Start with the Power Rule: Use it to transform terms with coefficients, such as converting \( 2 \log y \) into \( \log y^2 \).
• Apply Sum and Difference Rules: These rules combine the expression using one logarithm by applying the quotient rule and then the product rule, rearranging them into one single logarithmic expression, like \( \log(\frac{x z^3}{y^2}) \).

Understanding these rules helps you not only to solve exercises but also interpret real-world problems more effectively.

Power Rule for Logarithms

The power rule for logarithms is one of the key tools in simplifying expressions, especially those involving exponents within logarithms. The power rule states that \( n \log b = \log(b^n) \). This means if you have a product like \( n \cdot \log b \), it can be rewritten as a single logarithm with an exponent. For example, in the expression \( \log x - 2 \log y + 3 \log z \), we apply the power rule by rewriting it as \( \log x - \log y^2 + \log z^3 \).
The power rule is particularly useful because it enables the condensation of expressions by converting multiplication outside of the logarithm into an exponent inside it. This process lays the groundwork for further combining using the sum and difference rules, ultimately making the expression much simpler.