Problem 34 - College Algebra

Question

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{b}\left(\frac{\sqrt[3]{x} y^{4}}{z^{5}}\right) $$

Step-by-Step Solution

Answer

\(\log_b (\sqrt[3]{x} y^{4}/z^{5})\) can be expanded as \(1/3 \log_b(x) + 4 \log_b y - 5 \log_b(z)\)

Step 1: Identify the Properties of Logarithms Involved

The task involves three properties of logarithms. The first one is the quotient rule which states that \(\log_b (a/b) = \log_b a - \log_b b\). The second one is the product rule which states that \(\log_b (mn) = \log_b m + \log_b n\), and the third one is the power rule which states that \(\log_b a^n = n * \log_b a\).

Step 2: Apply the Quotient Rule

Applying the quotient rule to the expression \(\log_b ({\sqrt[3]{x} y^{4}}/{z^{5}})\), gives: \(\log_b(\sqrt[3]{x} y^{4}) - \log_b(z^{5})\)

Step 3: Apply the Product Rule

For the expression \(\log_b (\sqrt[3]{x} y^{4})\), apply the product rule as the inside has the form mn: \(\log_b (\sqrt[3]{x}) + \log_b y^{4}\)

Step 4: Apply the Power Rule

Now, apply the power rule to all three components: \(1/3 \log_b(x) + 4 \log_b y - 5 \log_b(z)\)