Question - Correct Evaluation of Logarithmic Expressions
Solution:
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.
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