Question

If $a,b,c\ne 0$ and belong to the set $\{0,1,2,3...9\}$, then ${\log }_{10}\left[\frac{(a+10b+{10}^{2}c)}{({10}^{-4}a+{10}^{-3}b+{10}^{-2}c)}\right]$

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is D

$4$

Find the value of $\mathit{l}\mathit{o}{\mathit{g}}_{\mathbf{10}}\left[\frac{(a+10b+{10}^{2}c)}{({10}^{-4}a+{10}^{-3}b+{10}^{-2}c)}\right]$:

$$\begin{gathered} ={\log }_{10}\left[\frac{(a+10b+{10}^{2}c)}{({10}^{-4}a+{10}^{-3}b+{10}^{-2}c)}\right] \\ ={\log }_{10}\left[\frac{(a+10b+{10}^{2}c)}{(a+10b+{10}^{2}c)}\times {10}^4\right] \\ ={\log }_{10}\left({10}^4\right) \\ =4 \end{gathered}$$

Hence, the correct option is (D).