Calculus Examples

$5 \log_{4} (x)$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$x^{5} = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

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

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

Step 1.2.2

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

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Step 1.2.2.1

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

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

Step 1.2.2.2

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.

$f (1) = 5 \log_{4} (1)$

Step 2.2

Simplify the result.

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Step 2.2.1

Logarithm base $4$ of $1$ is $0$ .

$f (1) = 5 \cdot 0$

Step 2.2.2

Multiply $5$ by $0$ .

$f (1) = 0$

Step 2.2.3

The final answer is $0$ .

$0$

Step 2.3

Convert $0$ to decimal.

$y = 0$

Step 3

Find the point at $x = 4$ .

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Step 3.1

Replace the variable $x$ with $4$ in the expression.

$f (4) = 5 \log_{4} (4)$

Step 3.2

Simplify the result.

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Step 3.2.1

Logarithm base $4$ of $4$ is $1$ .

$f (4) = 5 \cdot 1$

Step 3.2.2

Multiply $5$ by $1$ .

$f (4) = 5$

Step 3.2.3

The final answer is $5$ .

$5$

Step 3.3

Convert $5$ to decimal.

$y = 5$

Step 4

Find the point at $x = 2$ .

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Step 4.1

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

$f (2) = 5 \log_{4} (2)$

Step 4.2

Simplify the result.

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Step 4.2.1

Logarithm base $4$ of $2$ is $\frac{1}{2}$ .

$f (2) = 5 (\frac{1}{2})$

Step 4.2.2

Combine $5$ and $\frac{1}{2}$ .

$f (2) = \frac{5}{2}$

Step 4.2.3

The final answer is $\frac{5}{2}$ .

$\frac{5}{2}$

Step 4.3

Convert $\frac{5}{2}$ to decimal.

$y = 2.5$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 0), (4, 5), (2, 2.5)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 0 \\ 2 & 2.5 \\ 4 & 5 \end{array}$

Step 6