Problem 77 - 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, $$ \begin{aligned} \log _{b} \frac{x^{3}}{y^{2}} &=\log _{b} x^{3}-\log _{b} y^{2} \\ &=3 \log _{b} x-2 \log _{b} y \end{aligned} $$ $$ \log _{b} \sqrt[3]{x^{2} z} $$

Step-by-Step Solution

Answer

\( \log_b \sqrt[3]{x^2 z} = \frac{2}{3}\log_b x + \frac{1}{3}\log_b z \)

Step 1: Rewrite the expression with fractional exponents

The expression is given as \( \log_b \sqrt[3]{x^2 z} \). A cube root can be rewritten using fractional exponents: \( \sqrt[3]{x^2 z} = (x^2 z)^{1/3} \). Thus, the expression becomes \( \log_b (x^2 z)^{1/3} \).

Step 2: Apply the Power Rule for logarithms

The Power Rule states that \( \log_b (a^n) = n \cdot \log_b a \). Apply this rule to our expression: \( \log_b (x^2 z)^{1/3} = \frac{1}{3} \log_b (x^2 z) \).

Step 3: Apply the Product Rule for logarithms

The Product Rule for logarithms states that \( \log_b (ab) = \log_b a + \log_b b \). Here, apply it to the expression: \( \log_b (x^2 z) = \log_b x^2 + \log_b z \).

Step 4: Simplify using the Power Rule again

Now simplify \( \log_b x^2 \) using the Power Rule: \( \log_b x^2 = 2 \cdot \log_b x \). Substitute back into the equation: \( \log_b (x^2 z) = 2\log_b x + \log_b z \).

Step 5: Combine all parts

Substitute back into the expression from Step 2: \( \frac{1}{3}(2 \log_b x + \log_b z) \). This simplifies to \( \frac{2}{3}\log_b x + \frac{1}{3}\log_b z \).

Key Concepts

Fractional Exponents

Fractional exponents are a way to express roots as exponents. When you encounter a root expression such as a square root or a cube root, it can be rewritten using a fractional or rational exponent instead. For example, the square root of a number is the same as raising that number to the power of 1/2, and a cube root is like raising it to the power of 1/3. This concept is extremely useful in algebra and calculus because it allows us to apply the same exponent rules we use for regular powers.

In our problem, the expression \( \sqrt[3]{x^2 z} \) involves a cube root, which can be expressed using fractional exponents. So, it becomes \( (x^2 z)^{1/3} \). This transformation is crucial for applying power rules and product rules of logarithms later in simplifying the expression.

Power Rule for Logarithms

The Power Rule for logarithms is an essential tool when working with logarithmic expressions. This rule states that the logarithm of a number raised to an exponent is equal to the exponent times the logarithm of the number. In mathematical form: \( \log_b (a^n) = n \cdot \log_b a \).

This allows us to pull exponents out in front of the logarithm, simplifying expressions that include exponents inside the logarithmic terms. In the original exercise, we started with \( \log_b (x^2 z)^{1/3} \). By applying the Power Rule, we moved the fractional exponent \( \frac{1}{3} \) in front of the logarithm, resulting in the simpler form \( \frac{1}{3} \log_b (x^2 z) \). By doing so, we prepared the expression for further simplification with other rules.

Product Rule for Logarithms

The Product Rule for logarithms is another core principle when simplifying logarithmic expressions. This rule helps to break down logarithms of products into a sum of separate logarithms and is stated as \( \log_b (ab) = \log_b a + \log_b b \).

Using this rule, we can simplify complex logarithmic terms by expressing them as a sum. In the problem, we applied this rule to \( \log_b (x^2 z) \), meaning that \( \log_b (x^2 z) = \log_b x^2 + \log_b z \). This helps us separate each component within the argument of the logarithm, making it easier to apply additional rules such as the power rule again, if necessary.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions involves a systematic approach using several rules of logarithms, such as the power rule and product rule we discussed. By applying these rules, we can break down complicated logarithm expressions into simpler, more manageable forms.

After applying the Power Rule and the Product Rule, we took our expression \( \frac{1}{3} (2 \log_b x + \log_b z) \) further apart. First, \( \log_b x^2 \) was simplified using the Power Rule to become \( 2 \cdot \log_b x \). Then we apply the \( \frac{1}{3} \) coefficient from the initial fractional exponent to the entire simplified expression: \( \frac{2}{3} \log_b x + \frac{1}{3} \log_b z \).

This highlights the power of combining different logarithmic rules to achieve a form that is both exact and easier to handle mathematically. It reduces the expression to simpler parts and illustrates how logarithmic properties work together to unveil the expression's underlying simplicity.