Question & Answer
QUESTION

Express the following logarithms in terms of $\log a,\log b{\text{ and }}\log c$ .
$\log \left( {\sqrt[3]{{{a^2}}} \times \sqrt[2]{{{b^3}}}} \right)$.

ANSWER Verified Verified
Hint-In this question, we use the concept of properties of logarithm. We use property $\log \left( {xy} \right) = \log x + \log y{\text{ and }}\log \left( {{x^y}} \right) = y\log x$ . We have to express the question in terms of $\log a,\log b{\text{ and }}\log c$ so we use properties of logarithm mentioned in above.

Complete step-by-step answer:
Now, we have to solve problem $\log \left( {\sqrt[3]{{{a^2}}} \times \sqrt[2]{{{b^3}}}} \right)$ in term of $\log a,\log b{\text{ and }}\log c$.
We know term $\sqrt[3]{{{a^2}}}{\text{ and }}\sqrt[2]{{{b^3}}}$ are in product inside of logarithm. So, first we separate variables a and b by using property of logarithm,$\log \left( {xy} \right) = \log x + \log y$
$
   \Rightarrow \log \left( {\sqrt[3]{{{a^2}}} \times \sqrt[2]{{{b^3}}}} \right) \\
   \Rightarrow \log \left( {\sqrt[3]{{{a^2}}}} \right) + \log \left( {\sqrt[2]{{{b^3}}}} \right) \\
$
We can write as,
$ \Rightarrow \log \left( {{a^{\dfrac{2}{3}}}} \right) + \log \left( {{b^{\dfrac{3}{2}}}} \right)$
Now, we use another property $\log \left( {{x^y}} \right) = y\log x$
$ \Rightarrow \dfrac{2}{3}\log \left( a \right) + \dfrac{3}{2}\log \left( b \right)$
Now, we can see questions expressed in terms of $\log a,\log b{\text{ and }}\log c$.
So, the answer to the above problem is $\dfrac{2}{3}\log \left( a \right) + \dfrac{3}{2}\log \left( b \right)$ .
Note-In such types of problems we use some important points to solve questions in an easy way. First we separate variables a and b by using the property of logarithm $\log \left( {xy} \right) = \log x + \log y$ because we have to express questions in terms of loga and logb. Then we have to solve power and take it outside from the log by using another property $\log \left( {{x^y}} \right) = y\log x$ . So, we will get the required answer.