Graph f(x) = log base 1/3 of x-4

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

Find the asymptotes.

Set the argument of the logarithm equal to zero.

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

The vertical asymptote occurs at $x = 4$ .

Vertical Asymptote: $x = 4$

Find the point at $x = 5$ .

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

$f (5) = \log_{\frac{1}{3}} ((5) - 4)$

Simplify the result.

Subtract $4$ from $5$ .

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

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

$f (5) = 0$

The final answer is $0$ .

$0$

Convert $0$ to decimal.

$y = 0$

Find the point at $x = 7$ .

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

$f (7) = \log_{\frac{1}{3}} ((7) - 4)$

Simplify the result.

Subtract $4$ from $7$ .

$f (7) = \log_{\frac{1}{3}} (3)$

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

Rewrite as an equation.

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

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 $b^{y} = x$ .

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

Create expressions in the equation that all have equal bases.

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

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

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

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

$- x = 1$

Solve for $x$ .

$x = - 1$

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

$f (7) = - 1$

The final answer is $- 1$ .

$- 1$

Convert $- 1$ to decimal.

$y = - 1$

Find the point at $x = 13$ .

Replace the variable $x$ with $13$ in the expression.

$f (13) = \log_{\frac{1}{3}} ((13) - 4)$

Simplify the result.

Subtract $4$ from $13$ .

$f (13) = \log_{\frac{1}{3}} (9)$

Logarithm base $\frac{1}{3}$ of $9$ is $- 2$ .

Rewrite as an equation.

$\log_{\frac{1}{3}} (9) = x$

Rewrite $\log_{\frac{1}{3}} (9) = x$ 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 $b^{y} = x$ .

${(\frac{1}{3})}^{x} = 9$

Create expressions in the equation that all have equal bases.

${(3^{- 1})}^{x} = 3^{2}$

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

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

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

$- x = 2$

Solve for $x$ .

$x = - 2$

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

$f (13) = - 2$

The final answer is $- 2$ .

$- 2$

Convert $- 2$ to decimal.

$y = - 2$

The log function can be graphed using the vertical asymptote at $x = 4$ and the points $(5, 0), (7, - 1), (13, - 2)$ .

Vertical Asymptote: $x = 4$

$\begin{array}{c} x & y \\ 5 & 0 \\ 7 & - 1 \\ 13 & - 2 \end{array}$

Graph f(x) = log base 1/3 of x-4

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