Algebra Examples

$y = \log_{8} (x)$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$x = 0$

Step 1.2

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.

$f (1) = \log_{8} (1)$

Step 2.2

Simplify the result.

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Step 2.2.1

Logarithm base $8$ of $1$ is $0$ .

$f (1) = 0$

Step 2.2.2

The final answer is $0$ .

$0$

Step 2.3

Convert $0$ to decimal.

$y = 0$

Step 3

Find the point at $x = 8$ .

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Step 3.1

Replace the variable $x$ with $8$ in the expression.

$f (8) = \log_{8} (8)$

Step 3.2

Simplify the result.

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Step 3.2.1

Logarithm base $8$ of $8$ is $1$ .

$f (8) = 1$

Step 3.2.2

The final answer is $1$ .

$1$

Step 3.3

Convert $1$ to decimal.

$y = 1$

Step 4

Find the point at $x = 2$ .

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Step 4.1

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

$f (2) = \log_{8} (2)$

Step 4.2

Simplify the result.

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Step 4.2.1

Logarithm base $8$ of $2$ is $\frac{1}{3}$ .

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Step 4.2.1.1

Rewrite as an equation.

$\log_{8} (2) = x$

Step 4.2.1.2

Rewrite $\log_{8} (2) = x$ 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 $b^{y} = x$ .

$8^{x} = 2$

Step 4.2.1.3

Create expressions in the equation that all have equal bases.

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

Step 4.2.1.4

Rewrite ${(2^{3})}^{x}$ as $2^{3 x}$ .

$2^{3 x} = 2^{1}$

Step 4.2.1.5

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

$3 x = 1$

Step 4.2.1.6

Solve for $x$ .

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

Step 4.2.1.7

The variable $x$ is equal to $\frac{1}{3}$ .

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

Step 4.2.2

The final answer is $\frac{1}{3}$ .

$\frac{1}{3}$

Step 4.3

Convert $\frac{1}{3}$ to decimal.

$y = 0.\overline{3}$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 0), (8, 1), (2, 0.\overline{3})$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 0 \\ 2 & 0.333 \\ 8 & 1 \end{array}$

Step 6