Algebra Examples

$f (x) = - \frac{1}{2} \cdot \ln (x + 3)$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

${(x + 3)}^{\frac{1}{2}} = 0$

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.

$f (- 2) = - \frac{1}{2} \cdot \ln ((- 2) + 3)$

Step 2.2

Simplify the result.

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Step 2.2.1

Add $- 2$ and $3$ .

$f (- 2) = - \frac{1}{2} \cdot \ln (1)$

Step 2.2.2

The natural logarithm of $1$ is $0$ .

$f (- 2) = - \frac{1}{2} \cdot 0$

Step 2.2.3

Multiply $- \frac{1}{2} \cdot 0$ .

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Step 2.2.3.1

Multiply $0$ by $- 1$ .

$f (- 2) = 0 (\frac{1}{2})$

Step 2.2.3.2

Multiply $0$ by $\frac{1}{2}$ .

$f (- 2) = 0$

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 $x = - 1$ .

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Step 3.1

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

$f (- 1) = - \frac{1}{2} \cdot \ln ((- 1) + 3)$

Step 3.2

Simplify the result.

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Step 3.2.1

Add $- 1$ and $3$ .

$f (- 1) = - \frac{1}{2} \cdot \ln (2)$

Step 3.2.2

Multiply $- \frac{1}{2} \ln (2)$ .

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Step 3.2.2.1

Reorder $\ln (2)$ and $\frac{1}{2}$ .

$f (- 1) = - (\frac{1}{2} \cdot \ln (2))$

Step 3.2.2.2

Simplify $\frac{1}{2} \ln (2)$ by moving $\frac{1}{2}$ inside the logarithm.

$f (- 1) = - \ln (2^{\frac{1}{2}})$

Step 3.2.3

The final answer is $- \ln (2^{\frac{1}{2}})$ .

$- \ln (2^{\frac{1}{2}})$

Step 3.3

Convert $- \ln (2^{\frac{1}{2}})$ to decimal.

$y = - 0.34657359$

Step 4

Find the point at $x = 0$ .

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Step 4.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = - \frac{1}{2} \cdot \ln ((0) + 3)$

Step 4.2

Simplify the result.

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Step 4.2.1

Add $0$ and $3$ .

$f (0) = - \frac{1}{2} \cdot \ln (3)$

Step 4.2.2

Multiply $- \frac{1}{2} \ln (3)$ .

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Step 4.2.2.1

Reorder $\ln (3)$ and $\frac{1}{2}$ .

$f (0) = - (\frac{1}{2} \cdot \ln (3))$

Step 4.2.2.2

Simplify $\frac{1}{2} \ln (3)$ by moving $\frac{1}{2}$ inside the logarithm.

$f (0) = - \ln (3^{\frac{1}{2}})$

Step 4.2.3

The final answer is $- \ln (3^{\frac{1}{2}})$ .

$- \ln (3^{\frac{1}{2}})$

Step 4.3

Convert $- \ln (3^{\frac{1}{2}})$ to decimal.

$y = - 0.54930614$

Step 5

The log function can be graphed using the vertical asymptote at $x = - 3$ and the points $(- 2, 0), (- 1, - 0.34657359), (0, - 0.54930614)$ .

Vertical Asymptote: $x = - 3$

$\begin{array}{c} x & y \\ - 2 & 0 \\ - 1 & - 0.347 \\ 0 & - 0.549 \end{array}$

Step 6