Theory of Equations 1 Question 26
27. The number of solutions of $\log _4(x-1)=\log _2(x-3)$ is
(a) 3
(b) 1
(c) 2
(d) 0
(2001, 2M)
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Solution:
- Given, $\log _4(x-1)=\log _2(x-3)=\log _{4^{1 / 2}}(x-3)$
$$ \begin{array}{rlrl} \Rightarrow & \log _4(x-1) & =2 \log _4(x-3) \\ \Rightarrow & & \log _4(x-1) & =\log _4(x-3)^{2} \\ \Rightarrow & & (x-3)^{2} & =x-1 \\ \Rightarrow & x^{2}+9-6 x & =x-1 \\ \Rightarrow & & x^{2}-7 x+10=0 \\ \Rightarrow & & (x-2)(x-5)=0 \\ \Rightarrow & x=2, \text { or } x=5 \\ \Rightarrow & & x=5 \quad[\because x=2 \text { makes log }(x-3) \text { undefined }] . \end{array} $$
Hence, one solution exists.
