Trigonometry Examples

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

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

Solve for $x$ .

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Step 1.2.1

Subtract $3$ from both sides of the equation.

$- x = - 3$

Step 1.2.2

Divide each term in $- x = - 3$ by $- 1$ and simplify.

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Step 1.2.2.1

Divide each term in $- x = - 3$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 3}{- 1}$

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.

$\frac{x}{1} = \frac{- 3}{- 1}$

Step 1.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 3}{- 1}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $- 3$ by $- 1$ .

$x = 3$

Step 1.3

The vertical asymptote occurs at $x = 3$ .

Vertical Asymptote: $x = 3$

Step 2

Find the point at $x = 2$ .

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Step 2.1

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

$f (2) = \log_{\frac{1}{3}} (3 - (2))$

Step 2.2

Simplify the result.

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Step 2.2.1

Multiply $- 1$ by $2$ .

$f (2) = \log_{\frac{1}{3}} (3 - 2)$

Step 2.2.2

Subtract $2$ from $3$ .

$f (2) = \log_{\frac{1}{3}} (1)$

Step 2.2.3

Logarithm base $\frac{1}{3}$ of $1$ is $0$ .

$f (2) = 0$

Step 2.2.4

The final answer is $0$ .

$0$

Step 2.3

Convert $0$ to decimal.

$= 0$

Step 3

Find the point at $x = 1$ .

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Step 3.1

Replace the variable $x$ with $1$ in the expression.

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

Step 3.2

Simplify the result.

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Step 3.2.1

Multiply $- 1$ by $1$ .

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

Step 3.2.2

Subtract $1$ from $3$ .

$f (1) = \log_{\frac{1}{3}} (2)$

Step 3.2.3

The final answer is $\log_{\frac{1}{3}} (2)$ .

$\log_{\frac{1}{3}} (2)$

Step 3.3

Convert $\log_{\frac{1}{3}} (2)$ to decimal.

$= - 0.63092975$

Step 4

Find the point at $x = 0$ .

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Step 4.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = \log_{\frac{1}{3}} (3 - (0))$

Step 4.2

Simplify the result.

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Step 4.2.1

Subtract $0$ from $3$ .

$f (0) = \log_{\frac{1}{3}} (3)$

Step 4.2.2

Logarithm base $\frac{1}{3}$ of $3$ is $- 1$ .

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Step 4.2.2.1

Rewrite as an equation.

$\log_{\frac{1}{3}} (3) = x$

Step 4.2.2.2

Rewrite $\log_{\frac{1}{3}} (3) = x$ 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 $b^{y} = x$ .

${(\frac{1}{3})}^{x} = 3$

Step 4.2.2.3

Create expressions in the equation that all have equal bases.

${(3^{- 1})}^{x} = 3^{1}$

Step 4.2.2.4

Rewrite ${(3^{- 1})}^{x}$ as $3^{- x}$ .

$3^{- x} = 3^{1}$

Step 4.2.2.5

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

$- x = 1$

Step 4.2.2.6

Solve for $x$ .

$x = - 1$

Step 4.2.2.7

The variable $x$ is equal to $- 1$ .

$f (0) = - 1$

Step 4.2.3

The final answer is $- 1$ .

$- 1$

Step 4.3

Convert $- 1$ to decimal.

$= - 1$

Step 5

The log function can be graphed using the vertical asymptote at $x = 3$ and the points $(2, 0), (1, - 0.63092975), (0, - 1)$ .

Vertical Asymptote: $x = 3$

$\begin{array}{c} x & y \\ 0 & - 1 \\ 1 & - 0.631 \\ 2 & 0 \end{array}$

Step 6