Question

How do you condense $\dfrac{\ln 5}{2}+\dfrac{\ln 6}{2}+\dfrac{\ln 7}{2}$ ?

Answer 1

Hint: In this question the properties of logarithms are being used. The logarithm is the inverse function to exponential. That means that the inverse properties of all exponential properties are properties of logarithm. The properties that are being used in this question are:
$m{{\log }_{a}}b={{\log }_{a}}{{b}^{m}}....(i)$ and ${{\log }_{a}}b+{{\log }_{a}}c={{\log }_{a}}bc....(ii)$
First, we take $\dfrac{1}{2}$ common from the given expression and then use (ii) to combine the expression. Then use (i) to further simplify and it should give you the result as $\ln \left( {{210}^{\dfrac{1}{2}}} \right)$ . Then as we know that any number to the power on half is the square root we will have $\ln \left( \sqrt{210} \right)$ .

Complete step by step answer:
We have been asked to condense $\dfrac{\ln 5}{2}+\dfrac{\ln 6}{2}+\dfrac{\ln 7}{2}$ .
First, we take $\dfrac{1}{2}$ common from the expression, we have:
$\Rightarrow \dfrac{1}{2}\left( \ln 5+\ln 6+\ln 7 \right)$
Since $\ln x$ is nothing but${{\log }_{e}}x$,
We use the property of logarithms which says, ${{\log }_{a}}b+{{\log }_{a}}c={{\log }_{a}}bc....(ii)$, here $a=e$ , \[b=5\] and $c=6$ .
$\Rightarrow \dfrac{1}{2}\left( \ln 5+\ln 6+\ln 7 \right)=\dfrac{1}{2}\left( \ln \left( 5\times 6 \right)+\ln \left( 7 \right) \right)$
Using the same property again, we have $a=e$ , $b=5\times 6$ and $c=7$ .
$\Rightarrow \dfrac{1}{2}\left( \ln \left( 5\times 6 \right)+\ln \left( 7 \right) \right)=\dfrac{1}{2}\left( \ln \left( 5\times 6\times 7 \right) \right)$
Therefore, we have:
$\dfrac{\ln 5}{2}+\dfrac{\ln 6}{2}+\dfrac{\ln 7}{2}=\dfrac{1}{2}\left( \ln \left( 210 \right) \right)$
Now using $m{{\log }_{a}}b={{\log }_{a}}{{b}^{m}}....(i)$ property of logarithm, we have:
$\dfrac{1}{2}\left( \ln \left( 210 \right) \right)=\ln \left( {{210}^{\dfrac{1}{2}}} \right)$
As we know any number raised to the power $\dfrac{1}{2}$ is the square root of that number. So, we have:
$\ln \left( {{210}^{\dfrac{1}{2}}} \right)=\ln \left( \sqrt{210} \right)$

Thus, $\dfrac{\ln 5}{2}+\dfrac{\ln 6}{2}+\dfrac{\ln 7}{2}=\ln \left( \sqrt{210} \right)$ .

Note: One of the most common mistakes is to mis assume the property and take the sum inside log instead of taking the product. Another mistake could be when taking the half inside the log one might raise the base of the log to the power of half instead of raising the argument of log.
Alternatively, you do not have to take half common and then combine, you can directly take LCM to add them like fractions.