Trigonometry Examples

$f (x) = \log_{3} (3 x)$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$3 x = 0$

Step 1.2

Divide each term in $3 x = 0$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 x = 0$ by $3$ .

$\frac{3 x}{3} = \frac{0}{3}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{0}{3}$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{3}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $0$ by $3$ .

$x = 0$

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.

$f (1) = \log_{3} (3 (1))$

Step 2.2

Simplify the result.

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Step 2.2.1

Multiply $3$ by $1$ .

$f (1) = \log_{3} (3)$

Step 2.2.2

Logarithm base $3$ of $3$ is $1$ .

$f (1) = 1$

Step 2.2.3

The final answer is $1$ .

$1$

Step 2.3

Convert $1$ to decimal.

$y = 1$

Step 3

Find the point at $x = 3$ .

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Step 3.1

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

$f (3) = \log_{3} (3 (3))$

Step 3.2

Simplify the result.

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Step 3.2.1

Multiply $3$ by $3$ .

$f (3) = \log_{3} (9)$

Step 3.2.2

Logarithm base $3$ of $9$ is $2$ .

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Step 3.2.2.1

Rewrite as an equation.

$\log_{3} (9) = x$

Step 3.2.2.2

Rewrite $\log_{3} (9) = x$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b$ does not equal $1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$3^{x} = 9$

Step 3.2.2.3

Create equivalent expressions in the equation that all have equal bases.

$3^{x} = 3^{2}$

Step 3.2.2.4

Since the bases are the same, the two expressions are only equal if the exponents are also equal.

$x = 2$

Step 3.2.2.5

The variable $x$ is equal to $2$ .

$f (3) = 2$

Step 3.2.3

The final answer is $2$ .

$2$

Step 3.3

Convert $2$ to decimal.

$y = 2$

Step 4

Find the point at $x = 9$ .

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Step 4.1

Replace the variable $x$ with $9$ in the expression.

$f (9) = \log_{3} (3 (9))$

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply $3$ by $9$ .

$f (9) = \log_{3} (27)$

Step 4.2.2

Logarithm base $3$ of $27$ is $3$ .

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Step 4.2.2.1

Rewrite as an equation.

$\log_{3} (27) = x$

Step 4.2.2.2

Rewrite $\log_{3} (27) = x$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b$ does not equal $1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$3^{x} = 27$

Step 4.2.2.3

Create equivalent expressions in the equation that all have equal bases.

$3^{x} = 3^{3}$

Step 4.2.2.4

Since the bases are the same, the two expressions are only equal if the exponents are also equal.

$x = 3$

Step 4.2.2.5

The variable $x$ is equal to $3$ .

$f (9) = 3$

Step 4.2.3

The final answer is $3$ .

$3$

Step 4.3

Convert $3$ to decimal.

$y = 3$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 1), (3, 2), (9, 3)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 1 \\ 3 & 2 \\ 9 & 3 \end{array}$

Step 6