Problem 33 - Algebra and Trigonometry

Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\log _{b}\left(\frac{\sqrt{x} y^{3}}{z^{3}}\right)$$

Step-by-Step Solution

Answer

\(\frac{1}{2}\log_b(x) + 3 \log_b(y) - 3 \log_b(z)\)

Step 1: Applying the Quotient Property of Logarithms

The quotient property says that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator. So, we can write:\n\(\log _{b}\left(\frac{\sqrt{x} y^{3}}{z^{3}}\right) = \log_b\left(\sqrt{x} y^{3}\right) - \log_b(z^3)\)

Step 2: Using the Product Rule of logarithms

For the first term, we will use the product rule. According to the product rule, the logarithm of a product is equal to the sum of the logarithms of the individual factors. So, the expression becomes:\n\(\log_b\left(\sqrt{x} y^{3}\right) - \log_b(z^3) = \log_b(\sqrt{x}) + \log_b(y^3) - \log_b(z^3)\)

Step 3: Applying the Power property of Logarithms

For each term inside the logarithms, there's a power that we can pull out in front. The power property of logarithms states that \(\log_b(a^n) = n \log_b(a)\). So this yields:\n\(\frac{1}{2}\log_b(x) + 3 \log_b(y) - 3 \log_b(z)\)