Trigonometry Examples

$\log_{2} ({(x)}^{2}) + 2 \log_{2} (x) = 15$

Step 1

Move all the terms containing a logarithm to the left side of the equation.

$\log_{2} (x^{2}) + 2 \log_{2} (x) = 15$

Step 2

Simplify the left side.

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Step 2.1

Simplify $\log_{2} (x^{2}) + 2 \log_{2} (x)$ .

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Step 2.1.1

Simplify $2 \log_{2} (x)$ by moving $2$ inside the logarithm.

$\log_{2} (x^{2}) + \log_{2} (x^{2}) = 15$

Step 2.1.2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log_{2} (x^{2} x^{2}) = 15$

Step 2.1.3

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 2.1.3.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log_{2} (x^{2 + 2}) = 15$

Step 2.1.3.2

Add $2$ and $2$ .

$\log_{2} (x^{4}) = 15$

Step 3

Write in exponential form.

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Step 3.1

For logarithmic equations, $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ such that $x > 0$ , $b > 0$ , and $b \neq 1$ . In this case, $b = 2$ , $x = x^{4}$ , and $y = 15$ .

$b = 2$

$x = x^{4}$

$y = 15$

Step 3.2

Substitute the values of $b$ , $x$ , and $y$ into the equation $b^{y} = x$ .

$2^{15} = x^{4}$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $x^{4} = 2^{15}$ .

$x^{4} = 2^{15}$

Step 4.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt[4]{2^{15}}$

Step 4.3

Simplify $\pm \sqrt[4]{2^{15}}$ .

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Step 4.3.1

Raise $2$ to the power of $15$ .

$x = \pm \sqrt[4]{32768}$

Step 4.3.2

Rewrite $32768$ as $8^{4} \cdot 8$ .

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Step 4.3.2.1

Factor $4096$ out of $32768$ .

$x = \pm \sqrt[4]{4096 (8)}$

Step 4.3.2.2

Rewrite $4096$ as $8^{4}$ .

$x = \pm \sqrt[4]{8^{4} \cdot 8}$

Step 4.3.3

Pull terms out from under the radical.

$x = \pm 8 \sqrt[4]{8}$

Step 4.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 8 \sqrt[4]{8}$

Step 4.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 8 \sqrt[4]{8}$

Step 4.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 8 \sqrt[4]{8}, - 8 \sqrt[4]{8}$

Step 5

Exclude the solutions that do not make $\log_{2} ({(x)}^{2}) + 2 \log_{2} (x) = 15$ true.

$x = 8 \sqrt[4]{8}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = 8 \sqrt[4]{8}$

Decimal Form:

$x = 13.45434264 \dots$