Question

If ${\text{lo}}{{\text{g}}_{\text{2}}}{\text{(4 + lo}}{{\text{g}}_{\text{3}}}{\text{(x)) = 3}}$ , then the sum of digits of x isA. 3B. 6C. 9D. 10

Hint: As we know that ${{\text{a}}^{{\text{lo}}{{\text{g}}_{\text{a}}}{\text{(x)}}}}{\text{ = x}}$, we will apply the same in the given equation taking both sides of the expression exponent equivalent to the base of the logarithm function. Applying this method we’ll again be in the same situation so we’ll apply it again by taking both sides of the expression, exponent equivalent to the base of the logarithm function.

Given data: ${\text{lo}}{{\text{g}}_{\text{2}}}{\text{(4 + lo}}{{\text{g}}_{\text{3}}}{\text{(x)) = 3}}$
Now, solving for x in the equation ${\text{lo}}{{\text{g}}_{\text{2}}}{\text{(4 + lo}}{{\text{g}}_{\text{3}}}{\text{(x)) = 3}}$
It is well known that,
${{\text{a}}^{{\text{lo}}{{\text{g}}_{\text{a}}}{\text{(x)}}}}{\text{ = x}}$
Now, ${\text{lo}}{{\text{g}}_{\text{2}}}{\text{(4 + lo}}{{\text{g}}_{\text{3}}}{\text{(x)) = 3}}$
Taking both the sides as the exponent of 2, we get
${{\text{2}}^{{\text{lo}}{{\text{g}}_{\text{2}}}{\text{(4 + lo}}{{\text{g}}_{\text{3}}}{\text{(x))}}}}{\text{ = }}{{\text{2}}^{\text{3}}}$
Using ${{\text{a}}^{{\text{lo}}{{\text{g}}_{\text{a}}}{\text{(x)}}}}{\text{ = x}}$, we get,
$\Rightarrow {\text{4 + lo}}{{\text{g}}_{\text{3}}}{\text{(x) = 8}}$
On simplification we get,
$\Rightarrow {\text{lo}}{{\text{g}}_{\text{3}}}{\text{(x) = 8 - 4}} \\ \Rightarrow {\text{lo}}{{\text{g}}_{\text{3}}}{\text{(x) = 4}} \\$
Now, taking both the sides as the exponent of 3, we get
${{\text{3}}^{{\text{lo}}{{\text{g}}_{\text{3}}}{\text{(x)}}}}{\text{ = }}{{\text{3}}^{\text{4}}}$
Again, using ${{\text{a}}^{{\text{lo}}{{\text{g}}_{\text{a}}}{\text{(x)}}}}{\text{ = x}}$, we get,
${\text{x = }}{{\text{3}}^{\text{4}}} \\ \Rightarrow {\text{x = 81}} \\$
The digits in x i.e. 81 are 8 and 1, and their sum is 9.
Therefore, option (C) 9 is the correct option.

Note: An alternative method for doing this solution can be
It is well known that if
${\text{lo}}{{\text{g}}_{\text{x}}}{\text{y = a}}$ then,
${\text{y = }}{{\text{x}}^{\text{a}}}$
Applying this to the given equation, we’ll get
${\text{lo}}{{\text{g}}_{\text{2}}}{\text{(4 + lo}}{{\text{g}}_{\text{3}}}{\text{(x)) = 3}} \\ \Rightarrow {\text{4 + lo}}{{\text{g}}_{\text{3}}}{\text{(x) = }}{{\text{2}}^{\text{3}}} \\ \Rightarrow {\text{lo}}{{\text{g}}_{\text{3}}}{\text{(x) = 8 - 4}} \\ \Rightarrow {\text{lo}}{{\text{g}}_{\text{3}}}{\text{(x) = 4}} \\$
Again, applying the same formula
${\text{x = }}{{\text{3}}^{\text{4}}} \\ \Rightarrow {\text{x = 81}} \\$
Therefore the sum of digits in x is 9