Algebra Examples

$f (x) = 4^{x}$ and $g (x) = \log_{4} (x)$

Step 1

Exponential functions have a horizontal asymptote. The equation of the horizontal asymptote is $y = 0$ .

Horizontal Asymptote: $y = 0$

Step 2

Graph $g (x) = \log_{4} (x)$ .

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Step 2.1

Find the asymptotes.

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Step 2.1.1

Set the argument of the logarithm equal to zero.

$x = 0$

Step 2.1.2

The vertical asymptote occurs at $x = 0$ .

Vertical Asymptote: $x = 0$

Step 2.2

Find the point at $x = 1$ .

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Step 2.2.1

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

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

Step 2.2.2

Simplify the result.

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Step 2.2.2.1

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

$f (1) = 0$

Step 2.2.2.2

The final answer is $0$ .

$0$

Step 2.2.3

Convert $0$ to decimal.

$y = 0$

Step 2.3

Find the point at $x = 4$ .

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Step 2.3.1

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

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

Step 2.3.2

Simplify the result.

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Step 2.3.2.1

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

$f (4) = 1$

Step 2.3.2.2

The final answer is $1$ .

$1$

Step 2.3.3

Convert $1$ to decimal.

$y = 1$

Step 2.4

Find the point at $x = 2$ .

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Step 2.4.1

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

$f (2) = \log_{4} (2)$

Step 2.4.2

Simplify the result.

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Step 2.4.2.1

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

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Step 2.4.2.1.1

Rewrite as an equation.

$\log_{4} (2) = x$

Step 2.4.2.1.2

Rewrite $\log_{4} (2) = x$ 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 $b^{y} = x$ .

$4^{x} = 2$

Step 2.4.2.1.3

Create expressions in the equation that all have equal bases.

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

Step 2.4.2.1.4

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

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

Step 2.4.2.1.5

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

$2 x = 1$

Step 2.4.2.1.6

Solve for $x$ .

$x = \frac{1}{2}$

Step 2.4.2.1.7

The variable $x$ is equal to $\frac{1}{2}$ .

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

Step 2.4.2.2

The final answer is $\frac{1}{2}$ .

$\frac{1}{2}$

Step 2.4.3

Convert $\frac{1}{2}$ to decimal.

$y = 0.5$

Step 2.5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 0), (4, 1), (2, 0.5)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 0 \\ 2 & 0.5 \\ 4 & 1 \end{array}$

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 0 \\ 2 & 0.5 \\ 4 & 1 \end{array}$

Step 3

Plot each graph on the same coordinate system.

$f (x) = 4^{x}$

$g (x) = \log_{4} (x)$

Step 4