Trigonometry Examples

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

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

${(x + 3)}^{3} = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Set the $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 1.2.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 1.3

The vertical asymptote occurs at $x = - 3$ .

Vertical Asymptote: $x = - 3$

Step 2

Find the point at $x = - 2$ .

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Step 2.1

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

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

Step 2.2

Simplify the result.

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Step 2.2.1

Add $- 2$ and $3$ .

$f (- 2) = 3 \log (1)$

Step 2.2.2

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

$f (- 2) = 3 \cdot 0$

Step 2.2.3

Multiply $3$ by $0$ .

$f (- 2) = 0$

Step 2.2.4

The final answer is $0$ .

$0$

Step 2.3

Convert $0$ to decimal.

$y = 0$

Step 3

Find the point at $x = 7$ .

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Step 3.1

Replace the variable $x$ with $7$ in the expression.

$f (7) = 3 \log ((7) + 3)$

Step 3.2

Simplify the result.

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Step 3.2.1

Add $7$ and $3$ .

$f (7) = 3 \log (10)$

Step 3.2.2

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

$f (7) = 3 \cdot 1$

Step 3.2.3

Multiply $3$ by $1$ .

$f (7) = 3$

Step 3.2.4

The final answer is $3$ .

$3$

Step 3.3

Convert $3$ to decimal.

$y = 3$

Step 4

Find the point at $x = - 1$ .

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Step 4.1

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

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

Step 4.2

Simplify the result.

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Step 4.2.1

Add $- 1$ and $3$ .

$f (- 1) = 3 \log (2)$

Step 4.2.2

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

$f (- 1) = \log (2^{3})$

Step 4.2.3

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

$f (- 1) = \log (8)$

Step 4.2.4

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 $x = - 3$ and the points $(- 2, 0), (7, 3), (- 1, 0.90308998)$ .

Vertical Asymptote: $x = - 3$

$\begin{array}{c} x & y \\ - 2 & 0 \\ - 1 & 0.903 \\ 7 & 3 \end{array}$

Step 6