Trigonometry Examples

$y = - \log (x - 1)$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$x - 1 = 0$

Step 1.2

Add $1$ to both sides of the equation.

$x = 1$

Step 1.3

The vertical asymptote occurs at $x = 1$ .

Vertical Asymptote: $x = 1$

Step 2

Find the point at $x = 2$ .

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Step 2.1

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

$f (2) = - \log ((2) - 1)$

Step 2.2

Simplify the result.

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Step 2.2.1

Subtract $1$ from $2$ .

$f (2) = - \log (1)$

Step 2.2.2

Logarithm base $10$ of $1$ is $0$ .

$f (2) = - 0$

Step 2.2.3

Multiply $- 1$ by $0$ .

$f (2) = 0$

Step 2.2.4

The final answer is $0$ .

$0$

Step 2.3

Convert $0$ to decimal.

$y = 0$

Step 3

Find the point at $x = 11$ .

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Step 3.1

Replace the variable $x$ with $11$ in the expression.

$f (11) = - \log ((11) - 1)$

Step 3.2

Simplify the result.

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Step 3.2.1

Subtract $1$ from $11$ .

$f (11) = - \log (10)$

Step 3.2.2

Logarithm base $10$ of $10$ is $1$ .

$f (11) = - 1 \cdot 1$

Step 3.2.3

Multiply $- 1$ by $1$ .

$f (11) = - 1$

Step 3.2.4

The final answer is $- 1$ .

$- 1$

Step 3.3

Convert $- 1$ to decimal.

$y = - 1$

Step 4

Find the point at $x = 3$ .

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Step 4.1

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

$f (3) = - \log ((3) - 1)$

Step 4.2

Simplify the result.

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Step 4.2.1

Subtract $1$ from $3$ .

$f (3) = - \log (2)$

Step 4.2.2

The final answer is $- \log (2)$ .

$- \log (2)$

Step 4.3

Convert $- \log (2)$ to decimal.

$y = - 0.30102999$

Step 5

The log function can be graphed using the vertical asymptote at $x = 1$ and the points $(2, 0), (11, - 1), (3, - 0.30102999)$ .

Vertical Asymptote: $x = 1$

$\begin{array}{c} x & y \\ 2 & 0 \\ 3 & - 0.301 \\ 11 & - 1 \end{array}$

Step 6