Calculus Examples

$x^{5} \ln (x^{16})$

Step 1

Find the asymptotes.

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Set the argument of the logarithm equal to zero.

$x^{16} = 0$

Solve for $x$ .

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Take the 16th root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt[16]{0}$

Simplify $\pm \sqrt[16]{0}$ .

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Rewrite $0$ as $0^{16}$ .

$x = \pm \sqrt[16]{0^{16}}$

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

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

The vertical asymptote occurs at $x = 0$ .

Vertical Asymptote: $x = 0$

Step 2

Find the point at $x = 1$ .

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

$f (1) = {(1)}^{5} \ln ({(1)}^{16})$

Simplify the result.

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One to any power is one.

$f (1) = 1 \ln ({(1)}^{16})$

Multiply $\ln ({(1)}^{16})$ by $1$ .

$f (1) = \ln ({(1)}^{16})$

One to any power is one.

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

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

$f (1) = 0$

The final answer is $0$ .

$0$

Convert $0$ to decimal.

$y = 0$

Step 3

Find the point at $x = 2$ .

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

$f (2) = {(2)}^{5} \ln ({(2)}^{16})$

Simplify the result.

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Raise $2$ to the power of $5$ .

$f (2) = 32 \ln ({(2)}^{16})$

Raise $2$ to the power of $16$ .

$f (2) = 32 \ln (65536)$

Simplify $32 \ln (65536)$ by moving $32$ inside the logarithm.

$f (2) = \ln (65536^{32})$

The final answer is $\ln (65536^{32})$ .

$\ln (65536^{32})$

Convert $\ln (65536^{32})$ to decimal.

$y = 354.89135644$

Step 4

Find the point at $x = 3$ .

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

$f (3) = {(3)}^{5} \ln ({(3)}^{16})$

Simplify the result.

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Raise $3$ to the power of $5$ .

$f (3) = 243 \ln ({(3)}^{16})$

Raise $3$ to the power of $16$ .

$f (3) = 243 \ln (43046721)$

The final answer is $243 \ln (43046721)$ .

$243 \ln (43046721)$

Convert $243 \ln (43046721)$ to decimal.

$y = 4271.40457834$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 0), (2, 354.89135644), (3, 4271.40457834)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 0 \\ 2 & 354.891 \\ 3 & 4271.405 \end{array}$

Step 6