Problem 28 - 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. $$ \log _{b}\left(\frac{x^{3} y}{z^{2}}\right) $$

Step-by-Step Solution

Answer

The expanded form of \(\log _{b} \left( \frac{x^{3} y}{z^{2}} \right)\) is \(3 \log_b{x} + \log_b{y} - 2\log_b{z}\)

Step 1: Apply the Quotient Rule

The quotient rule of logarithms states that the logarithm of a quotient can be rewritten as a subtraction of the logarithms of the numerator and the denominator. So, apply this rule to the given expression: \(\log _{b}\left(\frac{x^{3} y}{z^{2}}\right) = \log_b{x^{3}y} - \log_b{z^{2}}\)

Step 2: Apply the Power and Product Rules

The power and product rules of logarithms state that the logarithm of a power can be rewritten as a product of the exponent and the logarithm of the base, and the logarithm of a product is the sum of the logarithms of the factors. Apply these rules in the simplified expression: \(3 \log_b{x} + \log_b{y} - 2\log_b{z}\)

Step 3: Final Simplified Form

Upon expanding and simplifying using the rules of logarithms, the final simplified expression is \(3 \log_b{x} + \log_b{y} - 2\log_b{z}\)