Problem 24 - 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. $$\log _{5}\left(\frac{\sqrt{x}}{25}\right)$$

Step-by-Step Solution

Answer

\(\frac{1}{2} * \log _{5} x - 2\)

Step 1: Apply Quotient Rule

The quotient rule for logarithms states that \(\log_b \frac{M}{N} = \log_b(M) - \log_b(N)\). Thus, the given expression \(\log _{5}\left(\frac{\sqrt{x}}{25}\right)\) can be rewritten as: \(\log _{5} \sqrt{x} - \log _{5} 25\).

Step 2: Apply Power Rule and Change of Base Formula

The power rule for logarithms states that \(\log_b M^n = n * \log_b(M)\). Similarly, the property of changing the base of the logarithm states that \(\log_b a = \log_c a / \log_c b, for any positive number c != 1.\) Thus, the above expression can be further simplified to: \(\frac{1}{2} * \log _{5} x - 2.\)

Step 3: Express Logarithm in Expanded Form

Applying the properties of logarithms, the expanded form of the provided logarithmic expression is thus: \(\frac{1}{2} * \log _{5} x - 2.\)