Question

If ${\log _{30}}3 = c$ and ${\log _{30}}5 = d$, then the value of ${\log _{30}}8$ is
$\left( a \right)$ 2(1 – c – d)
$\left( b \right)$3(1 + c + d)
$\left( c \right)$ 3(1 + c – d)
$\left( d \right)$ 3(1 – c – d)

Answer 1

Hint: In this particular type of question use the concept of logarithmic properties such that ${\log _a}{b^2} = 2{\log _a}b$, ${\log _b}\left( {\dfrac{m}{n}} \right) = {\log _b}m - {\log _b}n$, ${\log _b}b = 1$ and $\log \left( {mn} \right) = \log m + \log n$ so use these concepts to reach the solution of the given equation.

Complete step-by-step answer:
Given data,
${\log _{30}}3 = c$........................ (1)
And
${\log _{30}}5 = d$........................ (2)
Now we have to find the value of ${\log _{30}}8$
This equation is also written as, ${\log _{30}}{2^3}$
Now as we all know that according to logarithmic property the value of ${\log _a}{b^2} = 2{\log _a}b$ so use this property in the above equation we have,
So the given equation becomes,
$ \Rightarrow {\log _{30}}{2^3} = 3{\log _{30}}2$
Now 2 can be written as (30/15) so we have,
$ \Rightarrow {\log _{30}}{2^3} = 3{\log _{30}}\left( {\dfrac{{30}}{{15}}} \right)$
Now again according to logarithmic property the value of ${\log _b}\left( {\dfrac{m}{n}} \right) = {\log _b}m - {\log _b}n$ so use this property in the above equation we have,
\[ \Rightarrow {\log _{30}}{2^3} = 3\left( {{{\log }_{30}}30 - {{\log }_{30}}15} \right)\]
Now the above equation is also written as,
\[ \Rightarrow {\log _{30}}{2^3} = 3\left( {{{\log }_{30}}30 - {{\log }_{30}}\left( {3 \times 5} \right)} \right)\]
Now again according to logarithmic property the value of ${\log _b}b = 1$ and $\log \left( {mn} \right) = \log m + \log n$ so use these properties in the above equation we have,
\[ \Rightarrow {\log _{30}}{2^3} = 3\left( {1 - \left( {{{\log }_{30}}3 + {{\log }_{30}}5} \right)} \right)\]
Now from equation (1) and (2) we have,
\[ \Rightarrow {\log _{30}}{2^3} = 3\left( {1 - \left( {c + d} \right)} \right)\]
\[ \Rightarrow {\log _{30}}8 = 3\left( {1 - c - d} \right)\]
So this is the required answer.
Hence option (D) is the correct answer.

Note – Whenever we face such types of questions the key concept we have to remember is that always recall all the basic properties of log which is all stated above then simplify the given equation we want to solve according to these properties as above we will get the required answer.