Trigonometry Examples

$f (x) = - 3 \log (x)$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$x^{3} = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

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

$x = \sqrt[3]{0}$

Step 1.2.2

Simplify $\sqrt[3]{0}$ .

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Step 1.2.2.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 1.2.2.2

Pull terms out from under the radical, assuming real numbers.

$x = 0$

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.

$f (1) = - 3 \log (1)$

Step 2.2

Simplify the result.

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Step 2.2.1

Logarithm base $10$ of $1$ is $0$ .

$f (1) = - 3 \cdot 0$

Step 2.2.2

Multiply $- 3$ by $0$ .

$f (1) = 0$

Step 2.2.3

The final answer is $0$ .

$0$

Step 2.3

Convert $0$ to decimal.

$y = 0$

Step 3

Find the point at $x = 10$ .

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Step 3.1

Replace the variable $x$ with $10$ in the expression.

$f (10) = - 3 \log (10)$

Step 3.2

Simplify the result.

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Step 3.2.1

Logarithm base $10$ of $10$ is $1$ .

$f (10) = - 3 \cdot 1$

Step 3.2.2

Multiply $- 3$ by $1$ .

$f (10) = - 3$

Step 3.2.3

The final answer is $- 3$ .

$- 3$

Step 3.3

Convert $- 3$ to decimal.

$y = - 3$

Step 4

Find the point at $x = 2$ .

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Step 4.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = - 3 \log (2)$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify $- 3 \log (2)$ by moving $3$ inside the logarithm.

$f (2) = - \log (2^{3})$

Step 4.2.2

Raise $2$ to the power of $3$ .

$f (2) = - \log (8)$

Step 4.2.3

The final answer is $- \log (8)$ .

$- \log (8)$

Step 4.3

Convert $- \log (8)$ to decimal.

$y = - 0.90308998$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 0), (10, - 3), (2, - 0.90308998)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 0 \\ 2 & - 0.903 \\ 10 & - 3 \end{array}$

Step 6