Solve \log_2(2x^2) \cdot \log_2(16x) = \frac{9}{2}(\l

Question

Solve the equation

x_{1} = \frac{5 8}{2}, x_{2} = 16

Alternative Form

x_{1} \approx 0.757858, x_{2} = 16

Evaluate

lo g_{2} (2 x^{2}) \times lo g_{2} (16 x) = \frac{9}{2} (lo g_{2} (x))^{2}

Find the domain

More Steps

lo g_{2} (2 x^{2}) \times lo g_{2} (16 x) = \frac{9}{2} (lo g_{2} (x))^{2}, x > 0

Move the expression to the left side

lo g_{2} (2 x^{2}) \times lo g_{2} (16 x) - \frac{9}{2} (lo g_{2} (x))^{2} = 0

Use the logarithm product rule

4 + 9 lo g_{2} (x) - \frac{5}{2} (lo g_{2} (x))^{2} = 0

Solve the equation using substitution t = lo g_{2} (x)

4 + 9 t - \frac{5}{2} t^{2} = 0

Factor the expression

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\frac{1}{2} (4 - t) (2 + 5 t) = 0

Divide the terms

(4 - t) (2 + 5 t) = 0

When the product of factors equals 0,at least one factor is 0

4 - t = 0 2 + 5 t = 0

Solve the equation for t

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t = 4 2 + 5 t = 0

Solve the equation for t

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t = 4 t = - \frac{2}{5}

Substitute back

lo g_{2} (x) = 4 lo g_{2} (x) = - \frac{2}{5}

Calculate

More Steps

x = 16 lo g_{2} (x) = - \frac{2}{5}

Calculate

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x = 16 x = \frac{5 8}{2}

Check if the solution is in the defined range

x = 16 x = \frac{5 8}{2}, x > 0

Find the intersection of the solution and the defined range

x = 16 x = \frac{5 8}{2}

Solution

x_{1} = \frac{5 8}{2}, x_{2} = 16

Alternative Form

x_{1} \approx 0.757858, x_{2} = 16

Show Solution