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

Step-by-Step Solution

Answer

The expanded expression 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: Separation By Quotient Rule

Invoke the quotient rule for logarithms that states \(\log_b(\frac{m}{n}) = \log_b(m) - \log_b(n)\). Use this rule to separate the division inside the logarithm into two separate logarithms subtracted. Therefore, \(\log _{b}\left(\frac{x^{3} y}{z^{2}}\right)\) becomes \(\log_b(x^3 y) - \log_b(z^2)\).

Step 2: Apply the Product Rule

Invoke the product rule for logarithms that states \(\log_b(mn) = \log_b(m) + \log_b(n)\). Use this rule to separate the multiplication inside the first logarithm into two separate logarithms added. Therefore, \(\log_b(x^3 y) - \log_b(z^2)\) becomes \(\log_b(x^3) + \log_b(y) - \log_b(z^2)\).

Step 3: Applying the Power Rule

Invoke the power rule for logarithms that states \(\log_b(m^n) = n\log_b(m)\). Use this rule to bring the powers in the logarithmic argument out in front as coefficients. Therefore, \(\log_b(x^3) + \log_b(y) - \log_b(z^2)\) becomes \(3\log_b(x) + \log_b(y) - 2\log_b(z)\).