Calculus Examples

$\log (8 x)$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$8 x = 0$

Step 1.2

Divide each term in $8 x = 0$ by $8$ and simplify.

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Step 1.2.1

Divide each term in $8 x = 0$ by $8$ .

$\frac{8 x}{8} = \frac{0}{8}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $8$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{0}{8}$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{8}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $0$ by $8$ .

$x = 0$

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.

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

Step 2.2

Simplify the result.

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Step 2.2.1

Multiply $8$ by $1$ .

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

Step 2.2.2

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

$\log (8)$

Step 2.3

Convert $\log (8)$ to decimal.

$y = 0.90308998$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

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

Step 3.2

Simplify the result.

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Step 3.2.1

Multiply $8$ by $2$ .

$f (2) = \log (16)$

Step 3.2.2

The final answer is $\log (16)$ .

$\log (16)$

Step 3.3

Convert $\log (16)$ to decimal.

$y = 1.20411998$

Step 4

Find the point at $x = 3$ .

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Step 4.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = \log (8 (3))$

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply $8$ by $3$ .

$f (3) = \log (24)$

Step 4.2.2

The final answer is $\log (24)$ .

$\log (24)$

Step 4.3

Convert $\log (24)$ to decimal.

$y = 1.38021124$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 0.90308998), (2, 1.20411998), (3, 1.38021124)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 0.903 \\ 2 & 1.204 \\ 3 & 1.38 \end{array}$

Step 6