Question

If \[\log \left( {a + 1} \right) = \log \left( {4a - 3} \right) - \log 3\], then the value of a is:-
A) 3
B) 6
C) 8

Answer 1

Hint: Here we will use the property of logarithm function to solve the given problem and find the value of a.
The property of log function is:-
\[\log \left( {\dfrac{a}{b}} \right) = \log a - \log b\]

Complete step-by-step answer:
The given equation is:-
\[\log \left( {a + 1} \right) = \log \left( {4a - 3} \right) - \log 3\]
Now we know that according to the property of log function:
\[\log \left( {\dfrac{a}{b}} \right) = \log a - \log b\]
Hence applying this property in the given equation we get:-
\[\log \left( {a + 1} \right) = \log \left( {\dfrac{{4a - 3}}{3}} \right)\]
Now since we are getting log on both sides of the equation, we can equate the values inside the log function.
Hence on equating we get:-
\[a + 1 = \dfrac{{4a - 3}}{3}\]
Solving it further we get:-
\[3(a + 1) = 4a - 3\]
Simplifying it further we get:-
\[3a + 3 = 4a - 3\]
Solving for a we get:-
\[4a - 3a = 3 + 3\]
\[ \Rightarrow a = 6\]
Therefore, the value of a is 6

Therefore, option B is the correct option.

Note: The base of the log function cannot be negative.
The value inside the log function is always positive.
Students should take note that the value inside the log function cannot be equated until there is only a log function to be equated on both the sides.
If a scalar is multiplied on either side then first we need to change it into the power of the value inside the log function and then equate the two quantities.
$log(ab) = log(a) + log(b)$