Precalculus Examples

$f (x) = - 3 \log_{4} (x - 3)$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

${(x - 3)}^{3} = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Set the $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 1.2.2

Add $3$ to both sides of the equation.

$x = 3$

Step 1.3

The vertical asymptote occurs at $x = 3$ .

Vertical Asymptote: $x = 3$

Step 2

Find the point at $x = 4$ .

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Step 2.1

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

$f (4) = - 3 \log_{4} ((4) - 3)$

Step 2.2

Simplify the result.

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Step 2.2.1

Subtract $3$ from $4$ .

$f (4) = - 3 \log_{4} (1)$

Step 2.2.2

Logarithm base $4$ of $1$ is $0$ .

$f (4) = - 3 \cdot 0$

Step 2.2.3

Multiply $- 3$ by $0$ .

$f (4) = 0$

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 $x = 7$ .

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Step 3.1

Replace the variable $x$ with $7$ in the expression.

$f (7) = - 3 \log_{4} ((7) - 3)$

Step 3.2

Simplify the result.

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Step 3.2.1

Subtract $3$ from $7$ .

$f (7) = - 3 \log_{4} (4)$

Step 3.2.2

Logarithm base $4$ of $4$ is $1$ .

$f (7) = - 3 \cdot 1$

Step 3.2.3

Multiply $- 3$ by $1$ .

$f (7) = - 3$

Step 3.2.4

The final answer is $- 3$ .

$- 3$

Step 3.3

Convert $- 3$ to decimal.

$y = - 3$

Step 4

Find the point at $x = 5$ .

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Step 4.1

Replace the variable $x$ with $5$ in the expression.

$f (5) = - 3 \log_{4} ((5) - 3)$

Step 4.2

Simplify the result.

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Step 4.2.1

Subtract $3$ from $5$ .

$f (5) = - 3 \log_{4} (2)$

Step 4.2.2

Logarithm base $4$ of $2$ is $\frac{1}{2}$ .

$f (5) = - 3 (\frac{1}{2})$

Step 4.2.3

Combine $- 3$ and $\frac{1}{2}$ .

$f (5) = \frac{- 3}{2}$

Step 4.2.4

Move the negative in front of the fraction.

$f (5) = - \frac{3}{2}$

Step 4.2.5

The final answer is $- \frac{3}{2}$ .

$- \frac{3}{2}$

Step 4.3

Convert $- \frac{3}{2}$ to decimal.

$y = - 1.5$

Step 5

The log function can be graphed using the vertical asymptote at $x = 3$ and the points $(4, 0), (7, - 3), (5, - 1.5)$ .

Vertical Asymptote: $x = 3$

$\begin{array}{c} x & y \\ 4 & 0 \\ 5 & - 1.5 \\ 7 & - 3 \end{array}$

Step 6