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Domain of nested logarithms

This type of question, asked 4 times: 27 Jul 2022, 3 Apr 2025, 4 Apr 2025, 22 Jan 2026 S2. All 5 questions below, oldest first.

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  1. Question 1

    27 Jul 2022

    151
    The domain of the function f(x)=sin⁡−1[2x2−3]+log⁡2(log⁡12(x2−5x+5))f(x) =\sin^{- 1}\left\lbrack 2x^{2}- 3 \right\rbrack+\log_{2}\left( \log_{\frac{1}{2}}\left( x^{2}- 5x + 5 \right) \right), where [t]\lbrack t\rbrack is the greatest integer function, is:

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  2. Question 2

    3 Apr 2025

    67
    Let the domain of the function f(x)=log⁡2log⁡4log⁡6(3+4x−x2)f(x) =\log_{2}\log_{4}\log_{6}\left( 3 + 4x -x^{2} \right) be (a,b)(a,b). If ∫0b−a[x2]dx=p−q−r,p,q,r∈N,gcd(p,q,r)=1\int_{0}^{b-a} \left\lbrack x^{2} \right\rbrack dx=p-\sqrt{q}-\sqrt{r},p,q,r\in\mathbb{N},gcd(p,q,r) = 1, where [⋅]\lbrack \cdot \rbrack is the greatest integer function, then p+q+rp + q + r is equal to.

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  3. Question 3

    3 Apr 2025

    137
    If the domain of the function f(x)=log⁡7(1−log⁡4(x2−9x+18))f(x) =\log_{7}\left( 1 -\log_{4}\left( x^{2}- 9x+ 18 \right) \right) is (α,β)∪(γ,δ)(\alpha,\beta) \cup (\gamma,\delta), then α+β+γ+δ\alpha+\beta+\gamma+\delta is equal to.

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  4. Question 4

    4 Apr 2025

    133
    Let the domains of the functions f(x)=log⁡4log⁡3log⁡7(8−log⁡2(x2+4x+5))f(x) =\log_{4}\log_{3}\log_{7}\left( 8 -\log_{2}\left( x^{2}+ 4x+ 5 \right) \right) and g(x)=sin⁡−1(7x+10x−2)g(x) =\sin^{- 1}\left( \frac{7x+ 10}{x- 2} \right) be (α,β)(\alpha,\beta) and [γ,δ]\lbrack\gamma,\delta\rbrack, respectively. Then α2+β2+γ2+δ2\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2} is equal to :-

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  5. Question 5

    22 Jan 2026 S2

    64
    Let the domain of the function f(x)=log⁡3log⁡5(7−log⁡2(x2−10x+85))+sin⁡−1(∣3x−717−x∣)f(x) =\log_{3}\log_{5}\left( 7 -\log_{2}\left( x^{2}- 10x + 85 \right) \right)+\sin^{- 1}\left( \left| \frac{3x - 7}{17 - x} \right| \right) be (α,β](\alpha,\beta\rbrack. Then α+β\alpha + \beta is equal to :

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