Calculus Examples

$5 \ln (6 x)$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$7776 x^{5} = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Divide each term in $7776 x^{5} = 0$ by $7776$ and simplify.

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Step 1.2.1.1

Divide each term in $7776 x^{5} = 0$ by $7776$ .

$\frac{7776 x^{5}}{7776} = \frac{0}{7776}$

Step 1.2.1.2

Simplify the left side.

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Step 1.2.1.2.1

Cancel the common factor of $7776$ .

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Step 1.2.1.2.1.1

Cancel the common factor.

$\frac{7776 x^{5}}{7776} = \frac{0}{7776}$

Step 1.2.1.2.1.2

Divide $x^{5}$ by $1$ .

$x^{5} = \frac{0}{7776}$

Step 1.2.1.3

Simplify the right side.

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Step 1.2.1.3.1

Divide $0$ by $7776$ .

$x^{5} = 0$

Step 1.2.2

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

$x = \sqrt[5]{0}$

Step 1.2.3

Simplify $\sqrt[5]{0}$ .

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Step 1.2.3.1

Rewrite $0$ as $0^{5}$ .

$x = \sqrt[5]{0^{5}}$

Step 1.2.3.2

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.

$f (1) = 5 \ln (6 (1))$

Step 2.2

Simplify the result.

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Step 2.2.1

Multiply $6$ by $1$ .

$f (1) = 5 \ln (6)$

Step 2.2.2

Simplify $5 \ln (6)$ by moving $5$ inside the logarithm.

$f (1) = \ln (6^{5})$

Step 2.2.3

Raise $6$ to the power of $5$ .

$f (1) = \ln (7776)$

Step 2.2.4

The final answer is $\ln (7776)$ .

$\ln (7776)$

Step 2.3

Convert $\ln (7776)$ to decimal.

$y = 8.95879734$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

$f (2) = 5 \ln (6 (2))$

Step 3.2

Simplify the result.

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Step 3.2.1

Multiply $6$ by $2$ .

$f (2) = 5 \ln (12)$

Step 3.2.2

Simplify $5 \ln (12)$ by moving $5$ inside the logarithm.

$f (2) = \ln (12^{5})$

Step 3.2.3

Raise $12$ to the power of $5$ .

$f (2) = \ln (248832)$

Step 3.2.4

The final answer is $\ln (248832)$ .

$\ln (248832)$

Step 3.3

Convert $\ln (248832)$ to decimal.

$y = 12.42453324$

Step 4

Find the point at $x = 3$ .

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Step 4.1

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

$f (3) = 5 \ln (6 (3))$

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply $6$ by $3$ .

$f (3) = 5 \ln (18)$

Step 4.2.2

Simplify $5 \ln (18)$ by moving $5$ inside the logarithm.

$f (3) = \ln (18^{5})$

Step 4.2.3

Raise $18$ to the power of $5$ .

$f (3) = \ln (1889568)$

Step 4.2.4

The final answer is $\ln (1889568)$ .

$\ln (1889568)$

Step 4.3

Convert $\ln (1889568)$ to decimal.

$y = 14.45185878$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 8.95879734), (2, 12.42453324), (3, 14.45185878)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 8.959 \\ 2 & 12.425 \\ 3 & 14.452 \end{array}$

Step 6