Trigonometry Examples

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

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$3 x - 2 = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Add $2$ to both sides of the equation.

$3 x = 2$

Step 1.2.2

Divide each term in $3 x = 2$ by $3$ and simplify.

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Step 1.2.2.1

Divide each term in $3 x = 2$ by $3$ .

$\frac{3 x}{3} = \frac{2}{3}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{2}{3}$

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{3}$

Step 1.3

The vertical asymptote occurs at $x = \frac{2}{3}$ .

Vertical Asymptote: $x = \frac{2}{3}$

Step 2

Find the point at $x = 4$ .

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Step 2.1

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

$f (4) = \log (3 (4) - 2)$

Step 2.2

Simplify the result.

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Step 2.2.1

Multiply $3$ by $4$ .

$f (4) = \log (12 - 2)$

Step 2.2.2

Subtract $2$ from $12$ .

$f (4) = \log (10)$

Step 2.2.3

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

$f (4) = 1$

Step 2.2.4

The final answer is $1$ .

$1$

Step 2.3

Convert $1$ to decimal.

$y = 1$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

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

Step 3.2

Simplify the result.

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Step 3.2.1

Multiply $3$ by $2$ .

$f (2) = \log (6 - 2)$

Step 3.2.2

Subtract $2$ from $6$ .

$f (2) = \log (4)$

Step 3.2.3

The final answer is $\log (4)$ .

$\log (4)$

Step 3.3

Convert $\log (4)$ to decimal.

$y = 0.60205999$

Step 4

Find the point at $x = 3$ .

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Step 4.1

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

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

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply $3$ by $3$ .

$f (3) = \log (9 - 2)$

Step 4.2.2

Subtract $2$ from $9$ .

$f (3) = \log (7)$

Step 4.2.3

The final answer is $\log (7)$ .

$\log (7)$

Step 4.3

Convert $\log (7)$ to decimal.

$y = 0.84509804$

Step 5

The log function can be graphed using the vertical asymptote at $x = \frac{2}{3}$ and the points $(4, 1), (2, 0.60205999), (3, 0.84509804)$ .

Vertical Asymptote: $x = \frac{2}{3}$

$\begin{array}{c} x & y \\ 2 & 0.602 \\ 3 & 0.845 \\ 4 & 1 \end{array}$

Step 6