Question

If ${a^2} + 4{b^2} = 12ab$, then $\log \left( {a + 2b} \right)$ is
A. $\dfrac{1}{2}\left( {4\log 2 + \log a + \log b} \right)$
B. $0$
C. $\left( {\log 2 + \log a + \log b} \right)$
D. $\left( {4\log 2 + \log a - \log b} \right)$

Answer 1

Hint: At first, we will add $4ab$ on both the sides of equation ${a^2} + 4{b^2} = 12ab$. After this, we will apply $\log $ on both the sides of the equation and solve it further using logarithm formulas to get our required answer.

Formula Used:
Power rule: ${\log _b}\left( {{X^Y}} \right) = Y \times {\log _b}X$
Product rule: ${\log _b}\left( {X \times Y} \right) = {\log _b}X + {\log _b}Y$

Complete step by step solution:
 Given that, ${a^2} + 4{b^2} = 12ab$
Now, add $4ab$ on both the sides of the above written equation.
$ \Rightarrow {a^2} + 4{b^2} + 4ab = 12ab + 4ab$
We know that ${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$. Therefore, we get
$ \Rightarrow {\left( {a + 2b} \right)^2} = 16ab$
Now, take $\log $ on both the sides of the above written equation
$ \Rightarrow \log {\left( {a + 2b} \right)^2} = \log \left( {16ab} \right)$
We know that ${\log _b}\left( {{X^Y}} \right) = Y \times {\log _b}X$ and ${\log _b}\left( {X \times Y} \right) = {\log _b}X + {\log _b}Y$. Therefore, we get
$ \Rightarrow 2\log \left( {a + 2b} \right) = \log 16 + \log a + \log b$
$ \Rightarrow 2\log \left( {a + 2b} \right) = \log {2^4} + \log a + \log b$
Now, again apply ${\log _b}\left( {{X^Y}} \right) = Y \times {\log _b}X$
$ \Rightarrow 2\log \left( {a + 2b} \right) = 4\log 2 + \log a + \log b$
$ \Rightarrow \log \left( {a + 2b} \right) = \dfrac{1}{2}\left( {4\log 2 + \log a + \log b} \right)$
Hence, the value of $\log \left( {a + 2b} \right)$ is $\dfrac{1}{2}\left( {4\log 2 + \log a + \log b} \right)$.

Option ‘A’ is correct

Note: To solve this type of questions, one must know all the logarithmic rules. Some rules of the important logarithms are given below:
Product rule: the logarithm of the product is the total of the logarithm of the factors
${\log _b}\left( {X \times Y} \right) = {\log _b}X + {\log _b}Y$
Quotient rule: the logarithm of the ratio of two numbers is the difference between the logarithm of numerator and denominator
${\log _b}\left( {\dfrac{X}{Y}} \right) = {\log _b}X - {\log _b}Y$
Power rule: the logarithm of a positive number $X$ to power $Y$ is equivalent to the product of $Y$ and $\log $ of $X$
${\log _b}\left( {{X^Y}} \right) = Y \times {\log _b}X$