Question

which statement explains whether the equation is correct?
the equation is correct since $(log_{2}10)(log_{4}8)(log_{10}4)= \log(2 \cdot 10) \cdot \log(4 \cdot 8) \cdot \log(10 \cdot 4)\\= \log(20) \cdot \log(32) \cdot \log(40)\\= 3$
the equation is correct since $(log_{2}10)(log_{4}8)(log_{10}4)= \frac{\log10}{\log2} \cdot \frac{\log8}{\log4} \cdot \frac{\log4}{\log10}\\= \frac{\log8}{\log2}\\= 3$
the equation is not correct since $(log_{2}10)(log_{4}8)(log_{10}4)= \frac{\log10}{\log2} \cdot \frac{\log8}{\log4} \cdot \frac{\log4}{\log10}\\= \frac{\log8}{\log2}\\= 4$

Explanation:

Step1: Apply change of base formula

Recall the change of base formula: $\log_b a = \frac{\log a}{\log b}$. Apply it to each term:
$$(\log_{2}10)(\log_{4}8)(\log_{10}4) = \frac{\log 10}{\log 2} \cdot \frac{\log 8}{\log 4} \cdot \frac{\log 4}{\log 10}$$

Step2: Cancel common terms

Cancel $\log 10$ and $\log 4$ from numerator and denominator:
$$\frac{\log 10}{\log 2} \cdot \frac{\log 8}{\log 4} \cdot \frac{\log 4}{\log 10} = \frac{\log 8}{\log 2}$$

Step3: Simplify $\frac{\log 8}{\log 2}$

Rewrite $8$ as $2^3$, so $\log 8 = \log 2^3 = 3\log 2$:
$$\frac{\log 8}{\log 2} = \frac{3\log 2}{\log 2} = 3$$

Answer:

The equation is correct since $(\log_{2}10)(\log_{4}8)(\log_{10}4)=\frac{\log 10}{\log 2}\cdot\frac{\log 8}{\log 4}\cdot\frac{\log 4}{\log 10}=\frac{\log 8}{\log 2}=3$