Trigonometry Examples

$n = - 0.2 \log_{4} (v)$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$x^{0.2} = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Convert the decimal exponent to a fractional exponent.

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Step 1.2.1.1

Convert the decimal number to a fraction by placing the decimal number over a power of ten. Since there is $1$ number to the right of the decimal point, place the decimal number over $10^{1}$ $(10)$ . Next, add the whole number to the left of the decimal.

$x^{0 \frac{2}{10}} = 0$

Step 1.2.1.2

Reduce the fraction.

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Step 1.2.1.2.1

Convert $0 \frac{2}{10}$ to an improper fraction.

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Step 1.2.1.2.1.1

A mixed number is an addition of its whole and fractional parts.

$x^{0 + \frac{2}{10}} = 0$

Step 1.2.1.2.1.2

Add $0$ and $\frac{2}{10}$ .

$x^{\frac{2}{10}} = 0$

Step 1.2.1.2.2

Cancel the common factor of $2$ and $10$ .

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Step 1.2.1.2.2.1

Factor $2$ out of $2$ .

$x^{\frac{2 (1)}{10}} = 0$

Step 1.2.1.2.2.2

Cancel the common factors.

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Step 1.2.1.2.2.2.1

Factor $2$ out of $10$ .

$x^{\frac{2 \cdot 1}{2 \cdot 5}} = 0$

Step 1.2.1.2.2.2.2

Cancel the common factor.

$x^{\frac{2 \cdot 1}{2 \cdot 5}} = 0$

Step 1.2.1.2.2.2.3

Rewrite the expression.

$x^{\frac{1}{5}} = 0$

Step 1.2.2

Raise each side of the equation to the power of $\frac{1}{0.2}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{5}})}^{\frac{1}{0.2}} = 0^{\frac{1}{0.2}}$

Step 1.2.3

Simplify the exponent.

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Step 1.2.3.1

Simplify the left side.

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Step 1.2.3.1.1

Simplify ${(x^{\frac{1}{5}})}^{\frac{1}{0.2}}$ .

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Step 1.2.3.1.1.1

Multiply the exponents in ${(x^{\frac{1}{5}})}^{\frac{1}{0.2}}$ .

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Step 1.2.3.1.1.1.1

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

$x^{\frac{1}{5} \cdot \frac{1}{0.2}} = 0^{\frac{1}{0.2}}$

Step 1.2.3.1.1.1.2

Cancel the common factor of $0.2$ .

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Step 1.2.3.1.1.1.2.1

Factor $0.2$ out of $1$ .

$x^{\frac{0.2 (5)}{5} \cdot \frac{1}{0.2}} = 0^{\frac{1}{0.2}}$

Step 1.2.3.1.1.1.2.2

Cancel the common factor.

$x^{\frac{0.2 \cdot 5}{5} \cdot \frac{1}{0.2}} = 0^{\frac{1}{0.2}}$

Step 1.2.3.1.1.1.2.3

Rewrite the expression.

$x^{\frac{5}{5}} = 0^{\frac{1}{0.2}}$

Step 1.2.3.1.1.1.3

Divide $5$ by $5$ .

$x^{1} = 0^{\frac{1}{0.2}}$

Step 1.2.3.1.1.2

Simplify.

$x = 0^{\frac{1}{0.2}}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Simplify $0^{\frac{1}{0.2}}$ .

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Step 1.2.3.2.1.1

Divide $1$ by $0.2$ .

$x = 0^{5}$

Step 1.2.3.2.1.2

Raising $0$ to any positive power yields $0$ .

$x = 0$

Step 1.3

The vertical asymptote occurs at $x = 0$ .

Vertical Asymptote: $x = 0$

Step 2

Find the point at $x = 1$ .

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Step 2.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = - 0.2 \log_{4} (v)$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify $- 0.2 \log_{4} (v)$ by moving $0.2$ inside the logarithm.

$f (1) = - \log_{4} (v^{0.2})$

Step 2.2.2

The final answer is $- \log_{4} (v^{0.2})$ .

$- \log_{4} (v^{0.2})$

Step 3

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, - \log_{4} (v^{0.2})), (2, - \log_{4} (v^{0.2})), (3, - \log_{4} (v^{0.2}))$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & N a N \\ 2 & N a N \\ 3 & N a N \end{array}$

Step 4