Trigonometry Examples

$f (x) = \ln (3 - \ln (x))$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$3 - \ln (x) = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Subtract $3$ from both sides of the equation.

$- \ln (x) = - 3$

Step 1.2.2

Divide each term in $- \ln (x) = - 3$ by $- 1$ and simplify.

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Step 1.2.2.1

Divide each term in $- \ln (x) = - 3$ by $- 1$ .

$\frac{- \ln (x)}{- 1} = \frac{- 3}{- 1}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{\ln (x)}{1} = \frac{- 3}{- 1}$

Step 1.2.2.2.2

Divide $\ln (x)$ by $1$ .

$\ln (x) = \frac{- 3}{- 1}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $- 3$ by $- 1$ .

$\ln (x) = 3$

Step 1.2.3

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

$e^{\ln (x)} = e^{3}$

Step 1.2.4

Rewrite $\ln (x) = 3$ 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 $b^{y} = x$ .

$e^{3} = x$

Step 1.2.5

Rewrite the equation as $x = e^{3}$ .

$x = e^{3}$

Step 1.3

The vertical asymptote occurs at $x = e^{3}$ .

Vertical Asymptote: $x = e^{3}$

Step 2

Find the point at $x = 20$ .

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Step 2.1

Replace the variable $x$ with $20$ in the expression.

$f (20) = \ln (3 - \ln (20))$

Step 2.2

The final answer is $\ln (3 - \ln (20))$ .

$\ln (3 - \ln (20))$

Step 2.3

Convert $\ln (3 - \ln (20))$ to decimal.

$= - 5.45667404$

Step 3

Find the point at $x = 19$ .

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Step 3.1

Replace the variable $x$ with $19$ in the expression.

$f (19) = \ln (3 - \ln (19))$

Step 3.2

The final answer is $\ln (3 - \ln (19))$ .

$\ln (3 - \ln (19))$

Step 3.3

Convert $\ln (3 - \ln (19))$ to decimal.

$= - 2.89027338$

Step 4

Find the point at $x = 18$ .

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Step 4.1

Replace the variable $x$ with $18$ in the expression.

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

Step 4.2

The final answer is $\ln (3 - \ln (18))$ .

$\ln (3 - \ln (18))$

Step 4.3

Convert $\ln (3 - \ln (18))$ to decimal.

$= - 2.21066025$

Step 5

The log function can be graphed using the vertical asymptote at $x = e^{3}$ and the points $(20, - 5.45667404), (19, - 2.89027338), (18, - 2.21066025)$ .

Vertical Asymptote: $x = e^{3}$

$\begin{array}{c} x & y \\ 18 & - 2.211 \\ 19 & - 2.89 \\ 20 & - 5.457 \end{array}$

Step 6