Question

If \[\log (m+n)=\log m+\log n\], then:
A. \[mn=1\]
B. \[m=-n\]
C. \[\dfrac{m}{(m-1)}=n\]
D. \[\dfrac{m}{n}=1\]

Answer 1

Hint: In the given question, we are given an expression based on logarithm function. We have to use this given expression and find one of the options which is correct with respect to the given expression. We will be using the logarithm function property, which is, \[\log a+\log b=\log ab\]. We will apply this property in the given expression and solve the expression further until we get one of the forms as specified in the options.

Complete step by step solution:
According to the given question, we are given an expression which is based on logarithm function. We now have to use this given expression and find the option which is correct with respect to the given expression.
The expression we have is,
\[\log (m+n)=\log m+\log n\]----(1)
To go about modifying the given expression, we will have to use some of the properties of the logarithm function. We will be using the property, which is, \[\log a+\log b=\log ab\].
Applying this property in the equation (1), we will get,
\[\Rightarrow \log (m+n)=\log mn\]----(2)
In the equation (2), we can see that there is logarithm function across the equality, so we can equate their angles and we get,
\[\Rightarrow m+n=mn\]----(3)
We will now, re-arrange the equation (3) so that it looks like one of the given options. We have,
\[\Rightarrow m=mn-n\]----(4)
In equation (4), in the RHS side, we will take ‘n’ out as common, so we will get,
\[\Rightarrow m=n(m-1)\]
In the above expression, we will put the m terms on one side and n terms on the other side, we will get,
\[\Rightarrow \dfrac{m}{(m-1)}=n\]
Therefore, the answer is C. \[\dfrac{m}{(m-1)}=n\].

So, the correct answer is “Option C”.

Note: The logarithm property, \[\log a+\log b=\log ab\], is strictly applied to the RHS of the given logarithm expression. It was not required in the LHS side as the LHS had the m and n terms enclosed by the logarithm function. Also, while rearranging the expression keep an eye on the options given, so that you are not randomly re-arranging the given expression.