First slide

Types of real valued functions XI

Question

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

Moderate

A

all the roots are positive real numbers.
B

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

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

none of these

Solution

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

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