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Sagot :
To determine if the given equation
[tex]\[ \left(\log_2 10\right)\left(\log_4 8\right)\left(\log_{10} 4\right) = 3 \][/tex]
is correct, we will use the change of base formula, which states that for any positive numbers [tex]\(a, b,\)[/tex] and [tex]\(c\)[/tex],
[tex]\[ \log_a b = \frac{\log_c b}{\log_c a}. \][/tex]
### Step-by-Step Solution:
1. Calculate [tex]\(\log_2 10\)[/tex]:
Using the change of base formula,
[tex]\[ \log_2 10 = \frac{\log 10}{\log 2}. \][/tex]
2. Calculate [tex]\(\log_4 8\)[/tex]:
Similarly,
[tex]\[ \log_4 8 = \frac{\log 8}{\log 4}. \][/tex]
3. Calculate [tex]\(\log_{10} 4\)[/tex]:
Using the change of base formula again,
[tex]\[ \log_{10} 4 = \frac{\log 4}{\log 10}. \][/tex]
4. Multiply the expressions:
Now we multiply all the expressions together,
[tex]\[ \left(\log_2 10\right)\left(\log_4 8\right)\left(\log_{10} 4\right) = \left(\frac{\log 10}{\log 2}\right) \left(\frac{\log 8}{\log 4}\right) \left(\frac{\log 4}{\log 10}\right). \][/tex]
5. Simplify the product:
Notice that the logarithmic terms [tex]\(\log 10\)[/tex] and [tex]\(\log 4\)[/tex] will cancel out,
[tex]\[ \left(\frac{\log 10}{\log 2}\right) \left(\frac{\log 8}{\log 4}\right) \left(\frac{\log 4}{\log 10}\right) = \frac{\log 8}{\log 2}. \][/tex]
6. Evaluate the simplified expression:
Recall that [tex]\(8 = 2^3\)[/tex], so
[tex]\[ \log_2 8 = \log_2 (2^3) = 3 \log_2 2 = 3 \cdot 1 = 3. \][/tex]
So, we find that:
[tex]\[ \left(\log_2 10\right)\left(\log_4 8\right)\left(\log_{10} 4\right) = 3. \][/tex]
### Conclusion:
The correct statement that explains why the equation is correct is:
[tex]\[ \left(\log_2 10\right)\left(\log_4 8\right)\left(\log_{10} 4\right) = \frac{\log 10}{\log 2} \cdot \frac{\log 8}{\log 4} \cdot \frac{\log 4}{\log 10} = \frac{\log 8}{\log 2} = 3. \][/tex]
Thus, the equation is indeed correct. The only statement that correctly explains this is:
"The equation is correct since [tex]\(\left(\log_2 10\right)\left(\log_4 8\right)\left(\log_{10} 4\right) = \frac{\log 10}{\log 2} \cdot \frac{\log 8}{\log 4} \cdot \frac{\log 4}{\log 10} = \frac{\log 8}{\log 2} = 3\)[/tex]."
[tex]\[ \left(\log_2 10\right)\left(\log_4 8\right)\left(\log_{10} 4\right) = 3 \][/tex]
is correct, we will use the change of base formula, which states that for any positive numbers [tex]\(a, b,\)[/tex] and [tex]\(c\)[/tex],
[tex]\[ \log_a b = \frac{\log_c b}{\log_c a}. \][/tex]
### Step-by-Step Solution:
1. Calculate [tex]\(\log_2 10\)[/tex]:
Using the change of base formula,
[tex]\[ \log_2 10 = \frac{\log 10}{\log 2}. \][/tex]
2. Calculate [tex]\(\log_4 8\)[/tex]:
Similarly,
[tex]\[ \log_4 8 = \frac{\log 8}{\log 4}. \][/tex]
3. Calculate [tex]\(\log_{10} 4\)[/tex]:
Using the change of base formula again,
[tex]\[ \log_{10} 4 = \frac{\log 4}{\log 10}. \][/tex]
4. Multiply the expressions:
Now we multiply all the expressions together,
[tex]\[ \left(\log_2 10\right)\left(\log_4 8\right)\left(\log_{10} 4\right) = \left(\frac{\log 10}{\log 2}\right) \left(\frac{\log 8}{\log 4}\right) \left(\frac{\log 4}{\log 10}\right). \][/tex]
5. Simplify the product:
Notice that the logarithmic terms [tex]\(\log 10\)[/tex] and [tex]\(\log 4\)[/tex] will cancel out,
[tex]\[ \left(\frac{\log 10}{\log 2}\right) \left(\frac{\log 8}{\log 4}\right) \left(\frac{\log 4}{\log 10}\right) = \frac{\log 8}{\log 2}. \][/tex]
6. Evaluate the simplified expression:
Recall that [tex]\(8 = 2^3\)[/tex], so
[tex]\[ \log_2 8 = \log_2 (2^3) = 3 \log_2 2 = 3 \cdot 1 = 3. \][/tex]
So, we find that:
[tex]\[ \left(\log_2 10\right)\left(\log_4 8\right)\left(\log_{10} 4\right) = 3. \][/tex]
### Conclusion:
The correct statement that explains why the equation is correct is:
[tex]\[ \left(\log_2 10\right)\left(\log_4 8\right)\left(\log_{10} 4\right) = \frac{\log 10}{\log 2} \cdot \frac{\log 8}{\log 4} \cdot \frac{\log 4}{\log 10} = \frac{\log 8}{\log 2} = 3. \][/tex]
Thus, the equation is indeed correct. The only statement that correctly explains this is:
"The equation is correct since [tex]\(\left(\log_2 10\right)\left(\log_4 8\right)\left(\log_{10} 4\right) = \frac{\log 10}{\log 2} \cdot \frac{\log 8}{\log 4} \cdot \frac{\log 4}{\log 10} = \frac{\log 8}{\log 2} = 3\)[/tex]."
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