Correct Evaluation of Logarithmic Expressions
To solve this question, we need to evaluate the given logarithmic expressions and see which ones are correct. Here's an evaluation of each option:
A. \(\ln(1) = 0\)
This statement is true. The natural logarithm of 1 is always 0 because \(e^0 = 1\).
B. \(\log_2 9 = 3\)
This statement is false. The base 2 logarithm of 9 is not 3, because \(2^3 = 8\), not 9.
C. \(\log_{10} \frac{1}{100} = \frac{1}{2}\)
This statement is false. The base 10 logarithm of \(\frac{1}{100}\) is actually -2, because \(10^{-2} = \frac{1}{100}\).
D. \(\log_3 (-1) = \frac{1}{3}\)
This statement is false. The logarithm of a negative number is not a real number, and therefore, this expression cannot be evaluated in the real number system.
E. \(\log_5 \frac{1}{125} = -3\)
This statement is true. The base 5 logarithm of \(\frac{1}{125}\) is indeed -3, because \(5^{-3} = \frac{1}{125}\).
So, the correctly evaluated logarithmic expressions are A and E.
