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 calculate each logarithm individually and then multiply the results.
### Step-by-Step Calculation:
1. Calculate [tex]\(\log_2 10\)[/tex]:
By the change of base formula, we have:
[tex]\[
\log_2 10 = \frac{\log 10}{\log 2}
\][/tex]
Numerically, this value is approximately:
[tex]\[
\log_2 10 \approx 3.3219280948873626
\][/tex]
2. Calculate [tex]\(\log_4 8\)[/tex]:
By the change of base formula, we have:
[tex]\[
\log_4 8 = \frac{\log 8}{\log 4}
\][/tex]
Since [tex]\(8 = 2^3\)[/tex] and [tex]\(4 = 2^2\)[/tex], we can simplify [tex]\(\log_4 8\)[/tex] as:
[tex]\[
\log_4 8 = \frac{\log_2 8}{\log_2 4} = \frac{3}{2} = 1.5
\][/tex]
3. Calculate [tex]\(\log_{10} 4\)[/tex]:
By the change of base formula, we have:
[tex]\[
\log_{10} 4 = \frac{\log 4}{\log 10}
\][/tex]
Numerically, this value is approximately:
[tex]\[
\log_{10} 4 \approx 0.6020599913279623
\][/tex]
4. Multiply the results:
Now we multiply all the calculated values:
[tex]\[
(\log_2 10) \times (\log_4 8) \times (\log_{10} 4) \approx 3.3219280948873626 \times 1.5 \times 0.6020599913279623 = 3.0
\][/tex]
### Conclusion:
Thus, we find that:
[tex]\[
(\log_2 10) \times (\log_4 8) \times (\log_{10} 4) = 3.0
\][/tex]
Therefore, the equation is correct. This validates the statement as factually accurate.
The equation is correct since:
[tex]\[
(\log_2 10) \times (\log_4 8) \times (\log_{10} 4) = 3.0
\][/tex]
