Question

The value of $\log 5+\log 8-2\log 2$ is equal to:
(a). 1
(b). 8
(c). 4
(d). 2

Answer 1

Hint: The easiest way to approach this problem and simplify the given question is by using the logarithmic property $\log M+\log N=\log \left( MN \right)$ and $M\log N=\log {{N}^{M}}$ . Further, since the base of logarithm is not given, it is assumed to be 10.

Complete step-by-step answer:

There are four basic rules of logarithms as given below:-
${{\log }_{b}}\left( mn \right)={{\log }_{b}}m+{{\log }_{b}}n$
${{\log }_{b}}\left( \dfrac{m}{n} \right)={{\log }_{b}}m-{{\log }_{b}}n$
${{\log }_{b}}\left( {{m}^{n}} \right)=n{{\log }_{b}}m$
$lo{{g}_{b}}m=\dfrac{{{\log }_{a}}m}{{{\log }_{a}}b}$

Let us now consider the given question,

We know that, $\log M+\log N=\log \left( MN \right)$ and $M\log N=\log {{N}^{M}}$ .

Therefore, in case of this problem, we have,
log5 + log8 -2log2

Let us now use the property that ${{\log }_{b}}\left( mn \right)={{\log }_{b}}m+{{\log }_{b}}n$ to simplify $\log 5+\log 8$ and use ${{\log }_{b}}\left( {{m}^{n}} \right)=n{{\log }_{b}}m$ to simplify $2\log 2$

$=\log 5\times 8-\log {{2}^{2}}$
$=\log 40-\log 4$
Here we are using the property that ${{\log }_{b}}\left( \dfrac{m}{n} \right)={{\log }_{b}}m-{{\log }_{b}}n$ to simplify $\log 40-\log 4$
Thus, we have,
$=\log \dfrac{40}{4}$
$=\log 10$
We know that log 10 is actually ${{\log }_{10}}10$ .
Let us equate ${{\log }_{10}}10=n$
$\Rightarrow {{10}^{n}}=10$
$\Rightarrow n=1$

Hence, $\log 10=1$

Therefore, the final answer is option A.

Note: When we approach this question a lot of us make the mistake of thinking that we need to use the logarithmic tables to solve the question but, on the contrary, using logarithmic properties is much more efficient as it is useful for solving complex logarithmic problems as well.