Question

If ${{\log }_{a}}b=2$ ; ${{\log }_{b}}c=2$ and ${{\log }_{3}}c=3+{{\log }_{3}}a$ then (a+b+c) equals:
(a) 90
(b) 93
(c) 102
(d) None of these.

Answer 1

Hint: Start with the equation ${{\log }_{a}}b=2$ and use the definition of log, which will give $b={{a}^{2}}$ . Do the same with the second equation and use the first result to get the value of c in terms of a. Put the value of c you get in terms of a in the third given equation and solve to get the value of a. Once you get a, use the deduced results to get the value of b and c and solve.

Complete step by step solution:
Before starting with the solution to the above question we would have a look at the identities related to logarithmic functions:
${{\log }_{a}}x+{{\log }_{a}}y={{\log }_{a}}xy$
${{\log }_{a}}x-{{\log }_{a}}y={{\log }_{a}}\dfrac{x}{y}$
${{\log }_{{{a}^{b}}}}x=\dfrac{1}{b}{{\log }_{a}}x$
${{\log }_{a}}{{x}^{b}}=b{{\log }_{a}}x$
Now let us start the solution with the equation ${{\log }_{a}}b=2$ . We know ${{\log }_{y}}x=k\Rightarrow x={{y}^{k}}$ . So, if we use this in the equation, we get
${{\log }_{a}}b=2$
$\Rightarrow b={{a}^{2}}.........(i)$
Similarly, we will solve the equation ${{\log }_{b}}c=2$ .
${{\log }_{b}}c=2$
$\Rightarrow c={{b}^{2}}$
If we substitute the value of b from equation (i), we get
$c={{\left( {{a}^{2}} \right)}^{2}}={{a}^{4}}.........(ii)$
Now let us move to the equation ${{\log }_{3}}c=3+{{\log }_{3}}a$ and put the value of c. On doing so, we get
${{\log }_{3}}{{a}^{4}}=3+{{\log }_{3}}a$
We know ${{\log }_{a}}{{x}^{b}}=b{{\log }_{a}}x$ . If we use this in our equation, we get
$4{{\log }_{3}}a=3+{{\log }_{3}}a$
$\Rightarrow 3{{\log }_{3}}a=3$
$\Rightarrow {{\log }_{3}}a=1$
So, the value of logarithmic function is 1 if both the numbers in the logarithmic are equal.
$a=3$
If we put the value in equation (i), we get
$b={{a}^{2}}={{3}^{2}}=9$
Also, if we put a in equation (ii), we get
$c={{a}^{4}}={{3}^{4}}=81$
So, the value of (a+b+c) is equal to (3+9+81)=93.
Hence, the answer to the above question is option (b).

Note: Generally, the theory questions that we come across in our day to day life don’t need a logarithmic table to be solved; however, for solving the experimental data, you might require a logarithmic table. Also, it is very important that all the numbers in a logarithmic function including the base must be positive and should not be 1.