Trigonometry Examples

$10 \log_{9} (\sqrt[5]{y}) = 1$

Step 1

Subtract $1$ from both sides of the equation.

$10 \log_{9} (\sqrt[5]{y}) - 1 = 0$

Step 2

Simplify each term.

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Step 2.1

Simplify $10 \log_{9} (\sqrt[5]{y})$ by moving $10$ inside the logarithm.

$\log_{9} ({\sqrt[5]{y}}^{10}) - 1 = 0$

Step 2.2

Rewrite ${\sqrt[5]{y}}^{10}$ as $y^{2}$ .

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Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{y}$ as $y^{\frac{1}{5}}$ .

$\log_{9} ({(y^{\frac{1}{5}})}^{10}) - 1 = 0$

Step 2.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log_{9} (y^{\frac{1}{5} \cdot 10}) - 1 = 0$

Step 2.2.3

Combine $\frac{1}{5}$ and $10$ .

$\log_{9} (y^{\frac{10}{5}}) - 1 = 0$

Step 2.2.4

Cancel the common factor of $10$ and $5$ .

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Step 2.2.4.1

Factor $5$ out of $10$ .

$\log_{9} (y^{\frac{5 \cdot 2}{5}}) - 1 = 0$

Step 2.2.4.2

Cancel the common factors.

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Step 2.2.4.2.1

Factor $5$ out of $5$ .

$\log_{9} (y^{\frac{5 \cdot 2}{5 (1)}}) - 1 = 0$

Step 2.2.4.2.2

Cancel the common factor.

$\log_{9} (y^{\frac{5 \cdot 2}{5 \cdot 1}}) - 1 = 0$

Step 2.2.4.2.3

Rewrite the expression.

$\log_{9} (y^{\frac{2}{1}}) - 1 = 0$

Step 2.2.4.2.4

Divide $2$ by $1$ .

$\log_{9} (y^{2}) - 1 = 0$

Step 3

Set the argument in $\log_{9} (y^{2})$ less than or equal to $0$ to find where the expression is undefined.

$y^{2} \leq 0$

Step 4

Solve for $y$ .

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Step 4.1

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

$\sqrt{y^{2}} \leq \sqrt{0}$

Step 4.2

Simplify the equation.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Pull terms out from under the radical.

$| y | \leq \sqrt{0}$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Simplify $\sqrt{0}$ .

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Step 4.2.2.1.1

Rewrite $0$ as $0^{2}$ .

$| y | \leq \sqrt{0^{2}}$

Step 4.2.2.1.2

Pull terms out from under the radical.

$| y | \leq | 0 |$

Step 4.2.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$| y | \leq 0$

Step 4.3

Write $| y | \leq 0$ as a piecewise.

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Step 4.3.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$y \geq 0$

Step 4.3.2

In the piece where $y$ is non-negative, remove the absolute value.

$y \leq 0$

Step 4.3.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$y < 0$

Step 4.3.4

In the piece where $y$ is negative, remove the absolute value and multiply by $- 1$ .

$- y \leq 0$

Step 4.3.5

Write as a piecewise.

${\begin{cases} y \leq 0 & y \geq 0 \\ - y \leq 0 & y < 0 \end{cases}$

Step 4.4

Find the intersection of $y \leq 0$ and $y \geq 0$ .

$y = 0$

Step 4.5

Solve $- y \leq 0$ when $y < 0$ .

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Step 4.5.1

Divide each term in $- y \leq 0$ by $- 1$ and simplify.

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Step 4.5.1.1

Divide each term in $- y \leq 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y}{- 1} \geq \frac{0}{- 1}$

Step 4.5.1.2

Simplify the left side.

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Step 4.5.1.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} \geq \frac{0}{- 1}$

Step 4.5.1.2.2

Divide $y$ by $1$ .

$y \geq \frac{0}{- 1}$

Step 4.5.1.3

Simplify the right side.

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Step 4.5.1.3.1

Divide $0$ by $- 1$ .

$y \geq 0$

Step 4.5.2

Find the intersection of $y \geq 0$ and $y < 0$ .

No solution

Step 4.6

Find the union of the solutions.

$y = 0$

Step 5