If (x+1)log10⁡(x+1)=100(x+1) then

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$If (x + 1)^{\log_{10} (x + 1)} = 100 (x + 1) then$

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all the roots lie in the interval are positive real numbers.

all the roots lie in the interval (0, 100).

all the roots lie in the interval [-1, 99].

none of these.

answer is C.

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Detailed Solution

$(x + 1)^{\log_{10} (x + 1)} = 100 (x + 1)$

$\begin{array}{l} \Rightarrow \log_{10} (x + 1)^{\log_{10} (x + 1)} = \log_{10} (100 (x + 1)) \\ \log_{10} (x + 1) \log_{10} (x + 1) = 2 + \log_{10} (x + 1) \\ Let \log_{10} (x + 1) = y \\ \Rightarrow y^{2} - y - 2 = 0 \Rightarrow y = 2 or - 1 \Rightarrow \log_{10} (x + 1) = 2, - 1 \\ \Rightarrow x + 1 = 100, 1 / 10 \Rightarrow x = 99 or - 9 / 10 \end{array}$