Algebra Examples

y = \log_{3} (x)

$y = \log_{3} (x)$

Step 1

Find the asymptotes.

Tap for more steps...

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

$x = 1$ .

Tap for more steps...

Step 2.1

Replace the variable

x

$x$ with

1

$1$ in the expression.

f (1) = \log_{3} (1)

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

Step 2.2

Simplify the result.

Tap for more steps...

Step 2.2.1

Logarithm base

3

$3$ 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 = 3

$x = 3$ .

Tap for more steps...

Step 3.1

Replace the variable

x

$x$ with

3

$3$ in the expression.

f (3) = \log_{3} (3)

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

Step 3.2

Simplify the result.

Tap for more steps...

Step 3.2.1

Logarithm base

3

$3$ of

3

$3$ is

1

$1$ .

f (3) = 1

$f (3) = 1$

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 = 9

$x = 9$ .

Tap for more steps...

Step 4.1

Replace the variable

x

$x$ with

9

$9$ in the expression.

f (9) = \log_{3} (9)

$f (9) = \log_{3} (9)$

Step 4.2

Simplify the result.

Tap for more steps...

Step 4.2.1

Logarithm base

3

$3$ of

9

$9$ is

2

$2$ .

Tap for more steps...

Step 4.2.1.1

Rewrite as an equation.

\log_{3} (9) = x

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

Step 4.2.1.2

Rewrite

\log_{3} (9) = x

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

$b^{y} = x$ .

3^{x} = 9

$3^{x} = 9$

Step 4.2.1.3

Create equivalent expressions in the equation that all have equal bases.

3^{x} = 3^{2}

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

Step 4.2.1.4

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

x = 2

$x = 2$

Step 4.2.1.5

The variable

x

$x$ is equal to

2

$2$ .

f (9) = 2

$f (9) = 2$

f (9) = 2

$f (9) = 2$

Step 4.2.2

The final answer is

2

$2$ .

2

$2$

2

$2$

Step 4.3

Convert

2

$2$ to decimal.

y = 2

$y = 2$

y = 2

$y = 2$

Step 5

The log function can be graphed using the vertical asymptote at

x = 0

$x = 0$ and the points

(1, 0), (3, 1), (9, 2)

$(1, 0), (3, 1), (9, 2)$ .

Vertical Asymptote:

x = 0

$x = 0$

\begin{array}{c} x & y \\ 1 & 0 \\ 3 & 1 \\ 9 & 2 \end{array}

$\begin{array}{c} x & y \\ 1 & 0 \\ 3 & 1 \\ 9 & 2 \end{array}$

Step 6