Problem 35 - 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. $$\log _{5} \sqrt[3]{\frac{x^{2} y}{25}}$$

Step-by-Step Solution

Answer

The expanded form of \( \log_5 { \sqrt[3] { \frac {x^2 y}{25} } } \) is \( \frac {2}{3} \log_5 { x } + \frac {1}{3} \log_5 { y } - \frac {2}{3} \).

Step 1: Apply the Third Rule of Logarithms

The third rule of logarithms states that \( \log_b { a^m } = m \log_b { a }\). But before we can use it, notice that the inside of the log is a cube root, which can be rewritten with an exponent of \( \frac {1}{3} \). So \( \log_5 { \sqrt[3] { \frac {x^2 y}{25} } } \) can be rewritten as \( \frac {1}{3} \log_5 { \frac {x^2 y}{25} }\).

Step 2: Apply the First Rule of Logarithms

The first rule of logarithms states that \( \log_b { a \cdot c } = \log_b { a } + \log_b { c } \) and \( \log_b { a/c } = \log_b { a } - \log_b { c }\), where \( b, a, c > 0 \) and \( b \neq 1 \). So, the above expression can be further broken down as \( \frac {1}{3} (\log_5 { x^2 } + \log_5 { y } - \log_5 { 25 }) \).

Step 3: Apply the Third Rule of Logarithms Again

Apply the third rule of logarithms again to break down \( \log_5 { x^2 }\) as \( 2 \log_5 { x } \). Now the expression looks like this: \( \frac {1}{3} (2\log_5 { x } + \log_5 { y } - \log_5 { 25 }) \).

Step 4: Simplify the Expression

The last part to simplify is \( \log_5 { 25 }\). Since \( 5^2 \) is 25, \( \log_5 { 25 } = 2 \). Thus, substituting this into the equation gives the final expanded form of the logarithm: \( \frac {2}{3} \log_5 { x } + \frac {1}{3} \log_5 { y } - \frac {2}{3} \).