Problem 36 - College Algebra Essentials

Question

Evaluate each expression without using a calculator. $$\log _{11} 11$$

Step-by-Step Solution

Answer

The value of the expression \( \log _{11} 11 \) is 1.

Step 1: Conversion to Exponential Form

The logarithmic expression can be translated to exponential form using the rule \( \log_a b = c \) translates to \( a^c = b \). Applying this here, we get \( 11^{(\log _{11} 11)} = 11 \)

Step 2: Evaluate the exponent

Now, it should be easy to see what the exponent needs to be to make this equation true. As a definition, any number to the power of one equals the number itself. Therefore, for the equation \( 11^{(\log _{11} 11)} = 11 \) to be true, \( \log _{11} 11 \) must equal 1.