Question

If \[a,b,c \ne 0\] and belong to the set \[\left\{ {0,1,2,3,...9} \right\}\]. Then, what is the value of \[\log_{10}\left[ {\dfrac{{\left( {a + 10b + 10^{2}c} \right)}}{{\left( {10^{ - 4}a + 10^{ - 3}b + 10^{ - 2}c} \right)}}} \right]\]?
A. 1
B. 2
C. 3
D. 4

Answer 1

Hint In the given question, one logarithmic expression is given. Where the variables are belonged to the set \[\left\{ {0,1,2,3,...9} \right\}\]. First we will take common \[{10^{ - 4}}\] from the denominator and cancel out the common term. Then by using the logarithmic properties\[\log_{a}\left( {{a^m}} \right) = m\], we will find the value of \[\log_{10}\left[ {\dfrac{\left( {a + 10b + 10^2c} \right)}{\left( {10^{ - 4}a + 10^{ - 3}b + 10^{ - 2}c} \right)}} \right]\].

Formula used
\[\log_{a}\left( {{a^m}} \right) = m\]
Exponent rule: \[{a^m} \cdot {a^n} = {a^{m + n}}\]

Complete step by step solution:
The given logarithmic expression is \[\log_{10}\left[ {\dfrac{\left( {a + 10b + 10^2c} \right)}{\left( {10^{ - 4}a + 10^{ - 3}b + 10^{ - 2}c} \right)}} \right]\] where \[a,b,c \ne 0\].
Now simplify the above given expression.
\[L =\log_{10}\left[ {\dfrac{{\left( {a + 10b + 10^2}c \right)}}{{\left( {10^{ - 4}a + 10^{ - 3}b + 10^{ - 2}c} \right)}}} \right]\]
Take \[10^{ - 4}\] common from all terms of the denominator and use exponent rule.
\[ \Rightarrow \]\[L =\ log_{10}\left[ {\dfrac{{\left( {a + 10b + 10^{2}c} \right)}}{{10^{ - 4}\left( {a + 10b + 10^{2}c} \right)}}} \right]\]
\[ \Rightarrow \]\[L =\ log_{10}\left[ {\dfrac{1}{{10^{ - 4}}}} \right]\]
\[ \Rightarrow \]\[L =\ log_{10}\left[ {10^4} \right]\]
Now apply the logarithmic property \[\log_{a}\left( {{a^m}} \right) = m\].
\[L = 4\]
Hence the correct option is option D.

Note: There are two types of logarithm. The natural logarithm is as base \[e\]. The common logarithm is base 10. The given question is related to the common logarithm.