Solve log 2 (t-1) log 2 (2t-2) < 2

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve for t

- lo g_{2} (5) < t < 2 + lo g_{2} (5)

Alternative Form

t \in (- lo g_{2} (5), 2 + lo g_{2} (5))

Evaluate

lo g_{10} (2) \times (t - 1) \times lo g_{10} (2) \times (2 t - 2) < 2

Multiply the numbers

(lo g_{10} (2))^{2} (t - 1) (2 t - 2) < 2

Move the expression to the left side

(lo g_{10} (2))^{2} (t - 1) (2 t - 2) - 2 < 0

Expand the expression

More Steps

2 (lo g_{10} (2))^{2} t^{2} - 4 (lo g_{10} (2))^{2} t + 2 (lo g_{10} (2))^{2} - 2 < 0

Rewrite the expression

2 (lo g_{10} (2))^{2} t^{2} - 4 (lo g_{10} (2))^{2} t + 2 (lo g_{10} (2))^{2} - 2 = 0

Add or subtract both sides

2 (lo g_{10} (2))^{2} t^{2} - 4 (lo g_{10} (2))^{2} t = - 2 (lo g_{10} (2))^{2} + 2

Divide both sides

\frac{2 ( lo g _{10} ( 2 ) ) ^{2} t ^{2} - 4 ( lo g _{10} ( 2 ) ) ^{2} t}{2 ( lo g _{10} ( 2 ) ) ^{2}} = \frac{- 2 ( lo g _{10} ( 2 ) ) ^{2} + 2}{2 ( lo g _{10} ( 2 ) ) ^{2}}

Evaluate

t^{2} - 2 t = \frac{- ( lo g _{10} ( 2 ) ) ^{2} + 1}{( lo g _{10} ( 2 ) ) ^{2}}

Add the same value to both sides

t^{2} - 2 t + 1 = \frac{- ( lo g _{10} ( 2 ) ) ^{2} + 1}{( lo g _{10} ( 2 ) ) ^{2}} + 1

Simplify the expression

(t - 1)^{2} = \frac{1}{( lo g _{10} ( 2 ) ) ^{2}}

Take the root of both sides of the equation and remember to use both positive and negative roots

t - 1 = \pm \frac{1}{( lo g _{10} ( 2 ) ) ^{2}}

Simplify the expression

t - 1 = \pm \frac{1}{lo g _{10} ( 2 )}

Separate the equation into 2 possible cases

t - 1 = \frac{1}{lo g _{10} ( 2 )} t - 1 = - \frac{1}{lo g _{10} ( 2 )}

Solve the equation

More Steps

t = 2 + lo g_{2} (5) t - 1 = - \frac{1}{lo g _{10} ( 2 )}

Solve the equation

More Steps

t = 2 + lo g_{2} (5) t = - lo g_{2} (5)

Determine the test intervals using the critical values

t < - lo g_{2} (5) - lo g_{2} (5) < t < 2 + lo g_{2} (5) t > 2 + lo g_{2} (5)

Choose a value form each interval

t_{1} = - 3 t_{2} = 1 t_{3} = 5

To determine if t < - lo g_{2} (5) is the solution to the inequality,test if the chosen value t = - 3 satisfies the initial inequality

More Steps

t < - lo g_{2} (5) is not a solution t_{2} = 1 t_{3} = 5

To determine if - lo g_{2} (5) < t < 2 + lo g_{2} (5) is the solution to the inequality,test if the chosen value t = 1 satisfies the initial inequality

More Steps

t < - lo g_{2} (5) is not a solution - lo g_{2} (5) < t < 2 + lo g_{2} (5) is the solution t_{3} = 5

To determine if t > 2 + lo g_{2} (5) is the solution to the inequality,test if the chosen value t = 5 satisfies the initial inequality

More Steps

t < - lo g_{2} (5) is not a solution - lo g_{2} (5) < t < 2 + lo g_{2} (5) is the solution t > 2 + lo g_{2} (5) is not a solution

Solution

- lo g_{2} (5) < t < 2 + lo g_{2} (5)

Alternative Form

t \in (- lo g_{2} (5), 2 + lo g_{2} (5))

Show Solution