Question
Solve the inequality
x∈(2−10+3,0)∪(3,210+3)
Evaluate
log31(log41(x2−3x))<0
Find the domain
More Steps

Evaluate
{x2−3x>0log41(x2−3x)>0
Calculate
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Evaluate
x2−3x>0
Add the same value to both sides
x2−3x+49>49
Evaluate
(x−23)2>49
Take the 2-th root on both sides of the inequality
(x−23)2>49
Calculate
x−23>23
Separate the inequality into 2 possible cases
x−23>23x−23<−23
Calculate
x>3x−23<−23
Cancel equal terms on both sides of the expression
x>3x<0
Find the union
x∈(−∞,0)∪(3,+∞)
{x∈(−∞,0)∪(3,+∞)log41(x2−3x)>0
Calculate
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Evaluate
log41(x2−3x)>0
For 0<41<1 the expression log41(x2−3x)>0 is equivalent to x2−3x<(41)0
x2−3x<(41)0
Evaluate the power
x2−3x<1
Add the same value to both sides
x2−3x+49<1+49
Evaluate
x2−3x+49<413
Evaluate
(x−23)2<413
Take the 2-th root on both sides of the inequality
(x−23)2<413
Calculate
x−23<213
Separate the inequality into 2 possible cases
{x−23<213x−23>−213
Calculate
{x<213+3x−23>−213
Calculate
{x<213+3x>2−13+3
Find the intersection
2−13+3<x<213+3
{x∈(−∞,0)∪(3,+∞)2−13+3<x<213+3
Find the intersection
x∈(2−13+3,0)∪(3,213+3)
log31(log41(x2−3x))<0,x∈(2−13+3,0)∪(3,213+3)
For 0<31<1 the expression log31(log41(x2−3x))<0 is equivalent to log41(x2−3x)>(31)0
log41(x2−3x)>(31)0
Evaluate the power
log41(x2−3x)>1
For 0<41<1 the expression log41(x2−3x)>1 is equivalent to x2−3x<(41)1
x2−3x<(41)1
Evaluate the power
x2−3x<41
Add the same value to both sides
x2−3x+49<41+49
Evaluate
x2−3x+49<25
Evaluate
(x−23)2<25
Take the 2-th root on both sides of the inequality
(x−23)2<25
Calculate
x−23<210
Separate the inequality into 2 possible cases
{x−23<210x−23>−210
Calculate
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Evaluate
x−23<210
Move the constant to the right side
x<210+23
Write all numerators above the common denominator
x<210+3
{x<210+3x−23>−210
Calculate
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Evaluate
x−23>−210
Move the constant to the right side
x>−210+23
Write all numerators above the common denominator
x>2−10+3
{x<210+3x>2−10+3
Find the intersection
2−10+3<x<210+3
Check if the solution is in the defined range
2−10+3<x<210+3,x∈(2−13+3,0)∪(3,213+3)
Solution
x∈(2−10+3,0)∪(3,210+3)
Show Solution
