Math

QuestionSolve the inequality: log2x7logx+12<0\log ^{2} x - 7 \log x + 12 < 0.

Studdy Solution

STEP 1

Assumptions1. We are dealing with logarithms, so we assume that x>0x >0. . The base of the logarithms is not explicitly stated, so we assume that it is10.

STEP 2

First, we need to rewrite the inequality in a form that is easier to solve. We can do this by using the properties of logarithms to simplify the inequality.
log2x7logx+12<0\log ^{2} x-7 \log x+12<0This can be rewritten as(logx)27logx+12<0(\log x)^2 -7\log x +12 <0

STEP 3

Now, we can rewrite the inequality as a quadratic inequality by letting y=logxy = \log x.
(y)27y+12<0(y)^2 -7y +12 <0

STEP 4

We can solve the quadratic inequality by factoring it.
(y3)(y4)<0(y-3)(y-4) <0

STEP 5

Now, we can find the critical points of the inequality by setting each factor equal to zero.
y3=0y=3y-3 =0 \Rightarrow y =3y4=0y=4y-4 =0 \Rightarrow y =4

STEP 6

The solution to the inequality will be the intervals between the critical points where the inequality is true. We can find these intervals by testing a value from each interval in the inequality.
For y<3y <3, let's test y=2y =2(23)(24)=1>0(2-3)(2-4) =1 >0So the interval y<3y <3 is not part of the solution.
For 3<y<43 < y <4, let's test y=3.5y =3.5(3.53)(3.54)=0.25<0(3.5-3)(3.5-4) = -0.25 <0So the interval 3<y<43 < y <4 is part of the solution.
For y>4y >4, let's test y=5y =5(53)(54)=2>0(5-3)(5-4) =2 >0So the interval y>4y >4 is not part of the solution.

STEP 7

So the solution to the inequality in terms of yy is 3<y<43 < y <4.

STEP 8

Finally, we can convert the solution back to terms of xx by substituting y=logxy = \log x.
3<logx<43 < \log x <4This can be rewritten as103<x<10410^3 < x <10^4So the solution to the inequality log2x7logx+12<0\log ^{2} x-7 \log x+12<0 is 1000<x<100001000 < x <10000.

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