Solve log3(2-x) log31(x-1)>log333

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve for x

x \in \emptyset

Alternative Form

No solution

Evaluate

lo g_{10} (3) \times (2 - x) \times lo g_{10} (31) \times (x - 1) > lo g_{10} (333)

Multiply the terms

lo g_{10} (3) \times lo g_{10} (31) \times (2 - x) (x - 1) > lo g_{10} (333)

Move the expression to the left side

lo g_{10} (3) \times lo g_{10} (31) \times (2 - x) (x - 1) - lo g_{10} (333) > 0

Expand the expression

More Steps

3 lo g_{10} (3) \times lo g_{10} (31) \times x - 2 lo g_{10} (3) \times lo g_{10} (31) - lo g_{10} (3) \times lo g_{10} (31) \times x^{2} - lo g_{10} (333) > 0

Rewrite the expression

3 lo g_{10} (3) \times lo g_{10} (31) \times x - 2 lo g_{10} (3) \times lo g_{10} (31) - lo g_{10} (3) \times lo g_{10} (31) \times x^{2} - lo g_{10} (333) = 0

Simplify

3 lo g_{10} (3) \times lo g_{10} (31) \times x - 2 lo g_{10} (3) \times lo g_{10} (31) - lo g_{10} (333) - lo g_{10} (3) \times lo g_{10} (31) \times x^{2} = 0

Add or subtract both sides

3 lo g_{10} (3) \times lo g_{10} (31) \times x - lo g_{10} (3) \times lo g_{10} (31) \times x^{2} = 2 lo g_{10} (3) \times lo g_{10} (31) + lo g_{10} (333)

Divide both sides

\frac{3 lo g _{10} ( 3 ) \times lo g _{10} ( 31 ) \times x - lo g _{10} ( 3 ) \times lo g _{10} ( 31 ) \times x ^{2}}{- lo g _{10} ( 3 ) \times lo g _{10} ( 31 )} = \frac{2 lo g _{10} ( 3 ) \times lo g _{10} ( 31 ) + lo g _{10} ( 333 )}{- lo g _{10} ( 3 ) \times lo g _{10} ( 31 )}

Evaluate

- 3 x + x^{2} = - \frac{2 lo g _{10} ( 3 ) \times lo g _{10} ( 31 ) + lo g _{10} ( 333 )}{lo g _{10} ( 3 ) \times lo g _{10} ( 31 )}

Add the same value to both sides

- 3 x + x^{2} + \frac{9}{4} = - \frac{2 lo g _{10} ( 3 ) \times lo g _{10} ( 31 ) + lo g _{10} ( 333 )}{lo g _{10} ( 3 ) \times lo g _{10} ( 31 )} + \frac{9}{4}

Simplify the expression

(x - \frac{3}{2})^{2} = \frac{lo g _{3} ( \frac{3 1 ^{l o g_{10} (3)}}{33 3 ^{4}} )}{4 lo g _{10} ( 31 )}

Since the left-hand side is always positive or 0,and the right-hand side is always negative,the statement is false for any value of x

x \in / R

There are no key numbers,so choose any value to test,for example x = 0

x = 0

Solution

More Steps

x \in \emptyset

Alternative Form

No solution

Show Solution