Problem 80 - Intermediate Algebra

Question

For Problems \(69-80\), express each of the following as the sum or difference of simpler logarithmic quantities. Assume all variables represent positive real numbers. For example, $$ \begin{aligned} \log _{b} \frac{x^{3}}{y^{2}} &=\log _{b} x^{3}-\log _{b} y^{2} \\ &=3 \log _{b} x-2 \log _{b} y \end{aligned} $$ $$ \log _{b} \sqrt{\frac{x}{y}} $$

Step-by-Step Solution

Answer

\( \frac{1}{2} \log_{b} x - \frac{1}{2} \log_{b} y \)

Step 1: Rewrite the Square Root as an Exponent

The expression given is \( \log_{b} \sqrt{\frac{x}{y}} \). A square root can be written as an exponent of \( \frac{1}{2} \), so we can rewrite the expression as \( \log_{b} \left( \frac{x}{y} \right)^{\frac{1}{2}} \).

Step 2: Apply the Power Rule of Logarithms

Using the power rule \( \log_{b} a^{n} = n \cdot \log_{b} a \), we can move the exponent \( \frac{1}{2} \) in front of the logarithm. This transforms the expression into \( \frac{1}{2} \cdot \log_{b} \left( \frac{x}{y} \right) \).

Step 3: Use the Quotient Rule of Logarithms

The quotient rule states \( \log_{b} \frac{a}{c} = \log_{b} a - \log_{b} c \). Apply this rule to the expression \( \frac{1}{2} \cdot \log_{b} \left( \frac{x}{y} \right) \) to get \( \frac{1}{2} \left( \log_{b} x - \log_{b} y \right) \).

Step 4: Distribute the Factor

Finally, distribute the factor \( \frac{1}{2} \) across the expression inside the parentheses, resulting in \( \frac{1}{2} \cdot \log_{b} x - \frac{1}{2} \cdot \log_{b} y \). This represents the final simplified form.

Key Concepts

Power Rule of Logarithms

In the world of logarithms, the power rule is a handy tool that allows us to simplify expressions involving exponents. The rule states that for any positive real number **a** and any real number **n**, the logarithm of **a** raised to the power **n** can be expressed as the product of **n** and the logarithm of **a**. Mathematically, this is shown as:

• \( \log_{b} a^n = n \cdot \log_{b} a \)

This property is particularly useful when dealing with expressions where the variables are raised to a power, like in our exercise with \( \log_{b}\left(\frac{x}{y}\right)^{\frac{1}{2}} \).
You might wonder, why does this work? Essentially, logarithms were invented to transform multiplication into addition and powers into multiplication—making calculations more manageable.

In our example, applying the power rule allows us to "move" the exponent from inside the logarithm to the outside, simplifying the expression drastically. So, the operation:

• \( \log_{b}\left(\frac{x}{y}\right)^{\frac{1}{2}} \)
• becomes \( \frac{1}{2} \cdot \log_{b}\left(\frac{x}{y}\right) \).

Mastering this rule can significantly ease the process of dealing with complex logarithmic calculations.

Quotient Rule of Logarithms

This rule is like finding a shortcut on your math journey with logarithms. It helps us break down the logarithm of a division into a useful calculation. The **quotient rule** is expressed as:

• \( \log_{b} \frac{a}{c} = \log_{b} a - \log_{b} c \)

What this rule does is split the logarithm of a fraction into the difference between two separate logarithms. It’s especially helpful when you need to express logarithms simply.

In our given problem, we used the quotient rule after applying the power rule. Once we expressed the exponent:

• \( \frac{1}{2} \cdot \log_{b} \left( \frac{x}{y} \right) \)

The quotient rule then allows us to split this into:

• \( \frac{1}{2} \left( \log_{b} x - \log_{b} y \right) \)

This step pulls apart the fraction into separate, more manageable logarithmic terms that can be multiplied by \( \frac{1}{2} \), following properties of distribution.
A great tip to remember is that every time you see a division inside a logarithm, think about decomposing it using the quotient rule. This understanding will enhance your ability to navigate through complicated expressions with ease.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions is the art of transforming complex logarithms into a simpler, more digestible form. By using rules like the power and quotient rules, we can convert a tangled expression into a straightforward sum or difference of logs.

For example, let’s review our problem step-by-step:

• Start with \( \log_{b} \sqrt{\frac{x}{y}} \), represent the square root as an exponent of \( \frac{1}{2} \) .
• Convert it into \( \log_{b} \left( \frac{x}{y} \right)^{\frac{1}{2}} \).
• Apply the power rule: \( \frac{1}{2} \cdot \log_{b} \left( \frac{x}{y} \right) \).
• Then use the quotient rule: \( \frac{1}{2} \left( \log_{b} x - \log_{b} y \right) \).
• Distribute: \( \frac{1}{2} \cdot \log_{b} x - \frac{1}{2} \cdot \log_{b} y \).

Each step helps transition the expression into a clearer form as the rules simplify and break down the initial complex logarithm.

This method not only aids in managing equations during homework but also strengthens the understanding and manipulation of logarithms for tests and real-life applications, such as calculating compound interest or analyzing exponential growth scenarios. With practice, this simplification process becomes intuitive, making logarithmic problems less daunting to tackle.