Problem 24 - College Algebra

Question

In Exercises \(1-40,\) 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

The expanded and simplified form of the given logarithmic expression is \(\frac{1}{2} \log _{5} x - 2\).

Step 1: Applying the quotient rule

The quotient rule states that \(\log_b\frac{m}{n} = \log_b m - \log_b n\). Hence the given expression can be rewritten as: \[\log _{5}(\sqrt{x})-\log _{5}(25)\]

Step 2: Apply the power rule

The power rule states that \(\log_b m^n = n \log_b m\). Applying the power rule to our expression, we have: \[\frac{1}{2}\log _{5} x - \log _{5} (5^2)\]. Notice that, in the second part of the expression, we apply the power rule with a power of 2.

Step 3: Simplify the expression

We know that \(\log_b b^p = p\). So, \(\log _{5} 5^2 = 2\). Substitute this back into the expression to get the simplified version: \[\frac{1}{2}\log _{5} x - 2\]