Question

The number of solutions of $ {\log _4}(x - 1) = {\log _2}(x - 3) $ is?
A. 3
B. 1
C. 2
D. 0

Answer 1

Hint: Such functions are logarithmic functions as they have a form of $ f(x) = {\log _a}x, $ where a>0 and a is not equal to one. These functions are defined for all $ x \in (0,\infty ) $ and assume all real values. Therefore all the possible solutions of the above function will be real.

Complete step-by-step answer:
In the above question we know that the base of the logarithm as 4 can be written as the square of 2 therefore the base can be written as
 $ \Rightarrow {\log _{^{{2^2}}}}(x - 1) = {\log _2}(x - 3) $
We must know that the powers of the base can be rearranged by multiplying the reciprocal of that number whereas the power of the argument in the log is simply multiplied with the whole logarithmic function.
 $ \Rightarrow \dfrac{1}{2}{\log _2}(x - 1) = {\log _2}(x - 3) $
Multiplying with two on both the sides of the equation, We get,
  $ \Rightarrow {\log _2}(x - 1) = 2{\log _2}(x - 3) $
Therefore applying the above concept we observe that since the number is multiplied with the whole logarithmic function it can also be written as the power to the argument (x-3).
 $ {\log _2}(x - 1) = {\log _2}{(x - 3)^2} $
Now we can cancel the log on both the sides of the equation as the base is same in both the logarithmic function which is 2 to get a quadratic expression,
 $ x - 1 = {(x - 3)^2} $
Expanding the whole square at the R.H.S and rearranging all the terms at one side.
 $ \Rightarrow {x^2} - 6x + 9 - x + 1 = 0 $
 $ \Rightarrow {x^2} - 7x + 10 = 0 $
Solving the quadratic equation to get the value of x as its roots by the method of factorization, we get
 $ \Rightarrow (x - 5)(x - 2) = 0 $
 $ \Rightarrow x = 5 $ , $ x = 2 $
On putting the value of x=2 in the original equation, we see the log at R.H.S becomes $ \log ( - 1) $ which is not defined and hence the only solution is 5.
Therefore number of solutions = $ 1 $ .
So, the correct answer is “Option B”.

Note: In these logarithmic functions when the base is ‘e’ then the logarithmic function is called natural logarithmic function and when base is 10 then it is called a common logarithmic function.