Calculus Examples

$\ln (\ln (9 x))$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$\ln (9 x) = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (9 x)} = e^{0}$

Step 1.2.2

Rewrite $\ln (9 x) = 0$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{0} = 9 x$

Step 1.2.3

Solve for $x$ .

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Step 1.2.3.1

Rewrite the equation as $9 x = e^{0}$ .

$9 x = e^{0}$

Step 1.2.3.2

Anything raised to $0$ is $1$ .

$9 x = 1$

Step 1.2.3.3

Divide each term in $9 x = 1$ by $9$ and simplify.

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Step 1.2.3.3.1

Divide each term in $9 x = 1$ by $9$ .

$\frac{9 x}{9} = \frac{1}{9}$

Step 1.2.3.3.2

Simplify the left side.

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Step 1.2.3.3.2.1

Cancel the common factor of $9$ .

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Step 1.2.3.3.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{1}{9}$

Step 1.2.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{9}$

Step 1.3

The vertical asymptote occurs at $x = \frac{1}{9}$ .

Vertical Asymptote: $x = \frac{1}{9}$

Step 2

Find the point at $x = 2$ .

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Step 2.1

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

$f (2) = \ln (\ln (9 (2)))$

Step 2.2

Simplify the result.

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Step 2.2.1

Multiply $9$ by $2$ .

$f (2) = \ln (\ln (18))$

Step 2.2.2

The final answer is $\ln (\ln (18))$ .

$\ln (\ln (18))$

Step 2.3

Convert $\ln (\ln (18))$ to decimal.

$y = 1.06138512$

Step 3

Find the point at $x = 3$ .

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Step 3.1

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

$f (3) = \ln (\ln (9 (3)))$

Step 3.2

Simplify the result.

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Step 3.2.1

Multiply $9$ by $3$ .

$f (3) = \ln (\ln (27))$

Step 3.2.2

The final answer is $\ln (\ln (27))$ .

$\ln (\ln (27))$

Step 3.3

Convert $\ln (\ln (27))$ to decimal.

$y = 1.19266011$

Step 4

Find the point at $x = 4$ .

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Step 4.1

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

$f (4) = \ln (\ln (9 (4)))$

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply $9$ by $4$ .

$f (4) = \ln (\ln (36))$

Step 4.2.2

The final answer is $\ln (\ln (36))$ .

$\ln (\ln (36))$

Step 4.3

Convert $\ln (\ln (36))$ to decimal.

$y = 1.27634526$

Step 5

The log function can be graphed using the vertical asymptote at $x = \frac{1}{9}$ and the points $(2, 1.06138512), (3, 1.19266011), (4, 1.27634526)$ .

Vertical Asymptote: $x = \frac{1}{9}$

$\begin{array}{c} x & y \\ 2 & 1.061 \\ 3 & 1.193 \\ 4 & 1.276 \end{array}$

Step 6