Values of x satisfying the inequality x 1-log10x - log10x - 1 is -

The correct option is A x ϵ (0, 1) ∪ (1, 10^1/10)x^log_x10 . log_10x lt; 1 ⇒ 10 . log_10x lt; 1 ⇒log_10x lt; 1/10 ⇒ x lt; 10^1/10 and x gt; 0, ...

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Byju's Answer

Standard XII

Mathematics

Fundamental Laws of Logarithms

Values of x s...

Question

Values of $x$ satisfying the inequality $x^{(\frac{1}{{log}_{10} x})} . {log}_{10} x < 1$ is -

x ϵ (0, 10)

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x ϵ (0, 10^{10})

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x ϵ (0, 1) \cup (1, 10^{\frac{1}{10}})

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x ϵ (0, \frac{1}{10})

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Solution

The correct option is C $x ϵ (0, 1) \cup (1, 10^{\frac{1}{10}})$
$x^{{log}_{x} 10} . {log}_{10} x < 1$
$\Rightarrow 10 . {log}_{10} x < 1$
$\Rightarrow {log}_{10} x < \frac{1}{10}$
$\Rightarrow x < 10^{\frac{1}{10}}$
and $x > 0, x \neq 1$
So $0 < x < 10^{\frac{1}{10}}$ excluding $1.$

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Values of x satisfying the inequality x(1log10x).log10x<1 is -

The correct option is C x ϵ (0,1)∪(1,10110)xlogx10.log10x<1 ⇒10.log10x<1 ⇒log10x<110 ⇒x<10110 and x>0,x≠1 So 0<x<10110 excluding 1.

Values of $x$ satisfying the inequality $x^{(\frac{1}{{log}_{10} x})} . {log}_{10} x < 1$ is -

The correct option is C $x ϵ (0, 1) \cup (1, 10^{\frac{1}{10}})$
$x^{{log}_{x} 10} . {log}_{10} x < 1$
$\Rightarrow 10 . {log}_{10} x < 1$
$\Rightarrow {log}_{10} x < \frac{1}{10}$
$\Rightarrow x < 10^{\frac{1}{10}}$
and $x > 0, x \neq 1$
So $0 < x < 10^{\frac{1}{10}}$ excluding $1.$