Problem 29 - College Algebra

Question

Evaluate each expression without using a calculator. $$\log _{7} \sqrt{7}$$

Step-by-Step Solution

Answer

The result of the given expression is 0.5

Step 1: Rewrite the square root into a power

Since the square root of any number \(x\) can be represented as \(x^{1/2}\), the given expression can be rewritten as \(\log _{7} (7^{1/2})\).

Step 2: Apply the rule of logs

Now, apply the rule of logs that states \(\log _{a} (x^{y}) = y * \log _{a} (x)\). Thus, \(\log _{7} (7^{1/2}) = 1/2 * \log _{7} (7)\).

Step 3: Simplify further

Knowing that \(\log_{a}(a)\) is always equal to 1, we can simplify \(\log _{7} (7)\) to 1. This brings the expression to \(1/2 * 1\)

Step 4: Compute the final result

Finally, multiply 1/2 with 1, giving the result as 0.5