Algebra Examples

y = \log_{4} (x)

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

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

x = 0

$x = 0$

Step 1.2

The vertical asymptote occurs at

x = 0

$x = 0$ .

Vertical Asymptote:

x = 0

$x = 0$

Vertical Asymptote:

x = 0

$x = 0$

Step 2

Find the point at

x = 1

$x = 1$ .

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Step 2.1

Replace the variable

x

$x$ with

1

$1$ in the expression.

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

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

Step 2.2

Simplify the result.

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Step 2.2.1

Logarithm base

4

$4$ of

1

$1$ is

0

$0$ .

f (1) = 0

$f (1) = 0$

Step 2.2.2

The final answer is

0

$0$ .

0

$0$

0

$0$

Step 2.3

Convert

0

$0$ to decimal.

y = 0

$y = 0$

y = 0

$y = 0$

Step 3

Find the point at

x = 4

$x = 4$ .

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Step 3.1

Replace the variable

x

$x$ with

4

$4$ in the expression.

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

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

Step 3.2

Simplify the result.

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Step 3.2.1

Logarithm base

4

$4$ of

4

$4$ is

1

$1$ .

f (4) = 1

$f (4) = 1$

Step 3.2.2

The final answer is

1

$1$ .

1

$1$

1

$1$

Step 3.3

Convert

1

$1$ to decimal.

y = 1

$y = 1$

y = 1

$y = 1$

Step 4

Find the point at

x = 2

$x = 2$ .

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Step 4.1

Replace the variable

x

$x$ with

2

$2$ in the expression.

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

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

Step 4.2

Simplify the result.

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Step 4.2.1

Logarithm base

4

$4$ of

2

$2$ is

\frac{1}{2}

$\frac{1}{2}$ .

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Step 4.2.1.1

Rewrite as an equation.

\log_{4} (2) = x

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

Step 4.2.1.2

Rewrite

\log_{4} (2) = x

$\log_{4} (2) = x$ in exponential form using the definition of a logarithm. If

x

$x$ and

b

$b$ are positive real numbers and

b

$b$ does not equal

1

$1$ , then

\log_{b} (x) = y

$\log_{b} (x) = y$ is equivalent to

b^{y} = x

$b^{y} = x$ .

4^{x} = 2

$4^{x} = 2$

Step 4.2.1.3

Create expressions in the equation that all have equal bases.

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

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

Step 4.2.1.4

Rewrite

{(2^{2})}^{x}

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

2^{2 x}

$2^{2 x}$ .

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

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

Step 4.2.1.5

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

2 x = 1

$2 x = 1$

Step 4.2.1.6

Solve for

x

$x$ .

x = \frac{1}{2}

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

Step 4.2.1.7

The variable

x

$x$ is equal to

\frac{1}{2}

$\frac{1}{2}$ .

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

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

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

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

Step 4.2.2

The final answer is

\frac{1}{2}

$\frac{1}{2}$ .

\frac{1}{2}

$\frac{1}{2}$

\frac{1}{2}

$\frac{1}{2}$

Step 4.3

Convert

\frac{1}{2}

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

y = 0.5

$y = 0.5$

y = 0.5

$y = 0.5$

Step 5

The log function can be graphed using the vertical asymptote at

x = 0

$x = 0$ and the points

(1, 0), (4, 1), (2, 0.5)

$(1, 0), (4, 1), (2, 0.5)$ .

Vertical Asymptote:

x = 0

$x = 0$

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

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

Step 6