Example Question - logarithmic expressions evaluation
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Logarithmic Expressions Evaluation
The image displays four options with different logarithmic expressions evaluated to two decimal places. To check which one has been evaluated correctly, we would typically calculate the value of each logarithm and compare it to the given value. Here they are evaluated: A. log₃ 8 B. log₆ 6 C. log₄ 5 D. log₂ 32 Now, let's calculate these using base-10 logarithms (common logarithms) or base-e logarithms (natural logarithms) and the change of base formula: log_b(a) = log_k(a) / log_k(b) where k is any positive real number different from 1 (like 10 or e), and we'll use a calculator to find the common logarithm (base-10) or natural logarithm (base-e) of the numbers. A. log₃ 8 = log(8) / log(3) ≈ 0.90309 / 0.47712 ≈ 1.89 B. log₆ 6 = log(6) / log(6) ≈ 0.77815 / 0.77815 = 1 C. log₄ 5 = log(5) / log(4) ≈ 0.69897 / 0.60206 ≈ 1.16 D. log₂ 32 = log(32) / log(2) ≈ 1.50515 / 0.30103 ≈ 5 Now, let's round the results to the nearest hundredth and compare them to the supplied values: A. 1.89 does not match 0.43 B. 1 matches 1.63 (but note that the logarithm of a number to its own base equals 1 without the need for calculation) C. 1.16 matches 1.16 D. 5 does not match 1.51 As such, the only correctly evaluated expression to the nearest hundredth is: C. log₄ 5 = 1.16
Incorrect Evaluation of Logarithmic Expressions
The image shows a multiple-choice question asking which of the following logarithmic expressions have been evaluated correctly, to the nearest hundredth: A) \(\log_2 8 = 0.43\) B) \(\log_5 63 = 1.63\) C) \(\log_5 5 = 1.16\) D) \(\log_2 32 = 1.51\) Let's check each of the given logarithmic expressions one by one: A) \(\log_2 8\) The base-2 logarithm of 8 actually equals 3, because \(2^3 = 8\). So, this is incorrect. B) \(\log_5 63\) Using a calculator or logarithm tables to find the value of \(\log_5 63\) to two decimal places, you would see the actual value is around 2.80, not 1.63. So, this is incorrect. C) \(\log_5 5\) Given the fact that any log base itself is equal to 1 (\(\log_b b = 1\) for any b > 0), \(\log_5 5\) equals 1 exactly, not 1.16. So, this is incorrect. D) \(\log_2 32\) The base-2 logarithm of 32 actually equals 5, since \(2^5 = 32\). So, this is incorrect. None of the provided logarithmic expressions are evaluated correctly. Your next steps depend on the context in which this question was asked; it may be a matter of revisiting the definitions and properties of logarithms or reporting an issue with the question if it were intended that one of the answers be correct.
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.
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