Problem 27 - 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. $$\log _{b}\left(\frac{x^{2} y}{z^{2}}\right)$$

Step-by-Step Solution

Answer

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

Step 1: Apply Quotient Rule

Using the quotient rule which states that log(a/b) = log(a) - log(b), the given expression becomes:\n\n\[\log_b(x^2y) - \log_b(z^2)\]

Step 2: Apply Product Rule

Now, apply the product rule, which says that log(ab) = log(a) + log(b). The expression now is:\n\n\[\log_b(x^2) + \log_b(y) - \log_b(z^2)\].

Step 3: Apply Power Rule

Finally, apply the power rule, which states that log(a^n) = n log(a). The fully expanded expression becomes: \n\n\[2\log_b(x) + \log_b(y) - 2\log_b(z).\]