Logging inequalities 3

Algebra Level 2

$\large \log_{0.3}{(x-1)}<\log_{0.09}{(x-1)}$

Find the range of $x$ that satisfies the inequality above.

(-2,1)

(1,2)

(2,\infty)

None of these choices

1 solution

Mateo Matijasevick
Mar 8, 2016

$Note\quad that\quad 0.09={ (0.3) }^{ 2 }\\ So,\quad we\quad can\quad rewrite\quad the\quad equation\quad as:\quad \log _{ 0.3 }{ (x-1) } <\log _{ { (0.3 })^{ 2 } }{ (x-1) } \\ Which\quad is\quad equal\quad to:\quad \log _{ 0.3 }{ (x-1) } <\frac { 1 }{ 2 } \cdot \log _{ 0.3 }{ (x-1) } \\ Operating:\quad \log _{ 0.3 }{ (x-1) } <0\\ Changing\quad the\quad base:\quad \frac { \ln { (x-1) } }{ \ln { 0.3 } } <0.\quad As\quad \ln { 0.3 } <0,\quad the\quad inequality\quad changes:\quad \ln { (x-1) } >0\\ Finally,\quad solving\quad for\quad x:\quad x-1>1\\ Therefore,\quad x>2,\quad or\quad x\in (2,\infty )$

I chose none of the above as the statements are equal at x=2 and the question asks for less than, not less than or equal.

Owen Berendes - 5 years, 3 months ago

$(2,\infty)$ does not include 2. If it includes 2, the range would be written as $[2,\infty)$

Hung Woei Neoh - 5 years, 2 months ago

0.09=(0.3)^2 so we can write (x-1)^0.3=<(x-1) which is true for (x-1)>=1 so x>=2

damien G - 5 years, 2 months ago

Logging inequalities 3

1 solution

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