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Consider the following statements:
1. Solution of the inequality \( \Large log_{5} \left(x^{2}-11x+43\right)<2 \) is (0, 2)
2. If \( \Large [x - 1]^{ \left(log_{3}x^{2} - 2Logx^{9}\right)} =  \left(x - 1\right)^{7}  \), then x is 2 and 81.
Which of these is/are correct?


A) only(1)

B) only 2

C) both of these

D) none of these

Correct Answer:
B) only 2

Description for Correct answer:

(1) \( \Large alog_{5} \left(x^{2}-11x+43\right)<2 \)

\( \Large and x^{2}-11x+43>0 \)

=> \( \Large x^{2}-11x+43 < 5^{2}\)

\( \Large and \left(x - \frac{11}{2}\right)^{2}+\frac{51}{4}>0 \)

=> \( \Large x^{2}-11x+18<0 \)

\( \Large and \left(x-2\right) \left(x-9\right)<0 \)

\( \Large solution\ is\ \left(2, 9\right) \)



(2) \( \Large For\ domain |x, -1| ≠ 0, ≠ -1 \)

\( \Large Now |x-1| = 1, => x - 1 = \pm 1 \)

=> x = 0, 2

x = 0 is not in the domain and x = 2 satisfies the given equation.

If x-1 > 0 i.e., x > 1 then the given equation becomes

\( \Large 2 log_{3}x - \frac{4}{log_{3}x}=7 => x = 81, \frac{1}{\sqrt{3}} \)

\( \Large But \frac{1}{\sqrt{3}} \) being less than 1 is not valid

Hence, x = 2, 81


Part of solved Logarithms questions and answers : >> Elementary Mathematics >> Logarithms




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