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Logarithms

Section: Mathematics

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Question : 10 of 11

Marks: +1, -0

The number of solutions of the equation

\log_{4}(x-1)=\log_{2}(x-3)

is .......... .

[26 Feb 2021 Shift 1]

Your Answer:

Solution:

\log_{4}(x-1)=\log_{2}(x-3)

(given)

\Rightarrow \;\; \log_{2} 2(x-1)=\log_{2}(x-3)

Using property of logarithm,

\log_{b} c^{a}=\;\frac{1}{C}\log_{b} a

\Rightarrow \;\; \;\frac{1}{2}\log_{2}(x-1)=\log_{2}(x-3)

\Rightarrow \;\; \log_{2}(x-1)=2\log_{2}(x-3)

\Rightarrow \;\; \log_{2}(x-1)=\log_{2}(x-3)^{2}

On comparing,

x-1=(x-3)^{2}

\;\;\text{ or }\;\;x-1\;=x^{2}+9-6x

\Rightarrow\;x^{2}-7x+10\;=0

\Rightarrow\;x^{2}-5x-2x+10\;=0

\Rightarrow\;(x-5)(x-2)\;=0

\Rightarrow\;x\;=2,5

\;x\;=2\;\text{ (rejected) as }\;x>3

\therefore x=5

is only solution i.e. number of solution is 1 .

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