Problem 22 - 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. $$\log _{b}\left(x y^{3}\right)$$

Step-by-Step Solution

Answer

The expanded form of the given logarithmic expression is \( \log_b(x) + 3\log_b(y) \)

Step 1: Apply the product rule

Using the product rule of logarithms, we can write \( \log_b(xy^3) \) as \( \log_b(x) + \log_b(y^3) \). The product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors.

Step 2: Apply the power rule

Next, we can use the power rule of logarithms to further simplify \( \log_b(y^3) \). According to the power rule, we can move the exponent in the argument of the logarithm out front as a coefficient. Applying this rule gives us \( 3\log_b(y) \). Thus, our expression now is \( \log_b(x) + 3log_b(y) \).

Step 3: Final Expression

Combining the results from steps 1 and 2 gives a fully expanded form of the original logarithm as \( \log_b(x) + 3\log_b(y) \)