Problem 34 - Introductory and Intermediate Algebra for College Students 4th

Question

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

\(\frac{1}{3}\log_b x + 4*\log_b y - 5*\log_b z\)

Step 1: Apply Property of Division

Start by applying the property of logarithm that says \(\log_b (M/N) = \log_b M - \log_b N\). This means that we can rewrite the expression as: \(\log_b (\sqrt[3]{x} y^{4}) - \log_b (z^{5})\). Thus, we have decomposed the fraction inside the logarithm as a subtraction of two different logarithms.

Step 2: Apply Property of Multiplication

Next, let's use the second property which says \(\log_b (M*N) = \log_b M + \log_b N\). So the expression \(\log_b (\sqrt[3]{x} y^{4})\) can be rewritten as \(\log_b (\sqrt[3]{x}) + \log_b (y^{4})\).

Step 3: Apply Property of Exponents

The third step involves applying the power rule of logarithms, that \(\log_b (M^N) = N * \log_b M\). We can apply it to every individual logarithm we got after previous step. The expression thus becomes: \((1/3)\log_b x + 4*\log_b y - 5*\log_b z\).

Step 4: Evaluating the expression

Unfortunately, it is not possible to further evaluate the expression without knowledge of the values of x, y, z and b, or without a calculator. So the solution ends here.