Problem 29 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The value of \( \log _{7} \sqrt{7} \) is \(\frac{1}{2} \)

Step 1: Understand the Problem

Here, we need to evaluate the expression \( \log _{7} \sqrt{7} \) without using a calculator. The base of the logarithm is 7, and its argument is \(\sqrt{7}\), which is the same as \(7^{1/2}\). So, we are essentially trying to find the exponent that would get us from 7 (the base) to \( \sqrt{7} \) (the argument). The understanding of properties of logarithm is vital in carrying out this step.

Step 2: Apply the rule of logarithm

Having seen that our argument is a power, we can apply the rule of logarithm that allows us to move this exponent out front. In math terms, this can be described as: \( \log_b (a^n) = n \log_b (a) \). Hence, we can step down our expression \( \log _{7} \sqrt{7} \) to \( \frac{1}{2} \log _{7} 7 \) by bringing down the exponent 1/2, which is the square root.

Step 3: Evaluate the logarithm

The logarithm \( \log_{7} 7 \) implies a question: to what power should we raise 7 to obtain 7? Since anything raised to the power of 1 is itself, \(\log_{7} 7\) is equal to 1. Definitions like this are true for any logarithm, base and its argument being the same - it's always equal to 1. So our expression now simplifies to \( \frac{1}{2} \times 1 = \frac{1}{2} \)