To determine whether the equation [tex]\(\left(\log_2 10\right)\left(\log_4 8\right)\left(\log_{10} 4\right)=3\)[/tex] is correct, let's break down the problem step-by-step using logarithmic properties and the change of base formula.
1. Change of Base Formula:
The change of base formula states that [tex]\(\log_b a = \frac{\log_k a}{\log_k b}\)[/tex], where [tex]\(k\)[/tex] is any positive number different from 1.
Therefore:
[tex]\[
\log_2 10 = \frac{\log 10}{\log 2}
\][/tex]
[tex]\[
\log_4 8 = \frac{\log 8}{\log 4}
\][/tex]
[tex]\[
\log_{10} 4 = \frac{\log 4}{\log 10}
\][/tex]
2. Substitution in the Given Equation:
Substitute these into the equation:
[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]
3. Simplification:
Notice that [tex]\(\log 10\)[/tex] and [tex]\(\frac{1}{\log 10}\)[/tex] will cancel each other out when we multiply the fractions:
[tex]\[
\left(\frac{\log 10}{\log 2}\right) \left(\frac{\log 8}{\log 4}\right) \left(\frac{\log 4}{\log 10}\right) = \left(\frac{\log 8}{\log 2}\right)
\][/tex]
Now, let's simplify the term [tex]\(\frac{\log 8}{\log 2}\)[/tex]:
[tex]\[
\frac{\log 8}{\log 2} = \log_2 8
\][/tex]
4. Evaluate the Simplified Logarithm:
Since [tex]\(8\)[/tex] can be expressed as [tex]\(2^3\)[/tex], we have:
[tex]\[
\log_2 8 = \log_2 (2^3) = 3
\][/tex]
Therefore, the final value is:
[tex]\[
\left(\log_2 10\right)\left(\log_4 8\right)\left(\log_{10} 4\right) = 3
\][/tex]
So, the equation is indeed correct, and the correct statement explaining this is:
"The equation is correct since [tex]\(\left(\log_2 10\right)\left(\log_4 8\right)\left(\log_{10} 4\right) = \frac{\log 10}{\log 2} \cdot \frac{\log 8}{\log 4} \cdot \frac{\log 4}{\log 10}\)[/tex] [tex]\(\frac{\log 8}{\log 2} = 3\)[/tex]."
This statement confirms the correctness of the equation [tex]\(\left(\log_2 10\right)\left(\log_4 8\right)\left(\log_{10} 4\right) = 3\)[/tex].
