log-x-x-2-7-1-

Question Number 207559 by hardmath last updated on 18/May/24

\log_{x} (x^{2} + 7) ⩽ 1

Answered by mr W last updated on 19/May/24

\log_{x} (x^{2} + 7) - \log_{x} x ⩽ 0

\log_{x} (\frac{x^{2} + 7}{x}) ⩽ 0

\frac{\log (\frac{x^{2} + 7}{x})}{\log x} ⩽ 0

0 < x < 1 \land \frac{x^{2} + 7}{x} ⩾ 1

0 < x < 1 \land {(x - \frac{1}{2})}^{2} + \frac{27}{4} ⩾ 0

\Rightarrow 0 < x < 1 ✓

o r

1 < x \land 0 < \frac{x^{2} + 7}{x} ⩽ 1

1 < x \land {(x - \frac{1}{2})}^{2} + \frac{27}{4} ⩽ 0

\Rightarrow n o s o l u t i o n!

Commented by hardmath last updated on 19/May/24

thank you professor