Pre-Algebra Examples

$\ln (3 x + 1)$

Step 1

Find the asymptotes.

Tap for more steps...

Step 1.1

Set the argument of the logarithm equal to zero.

$3 x + 1 = 0$

Step 1.2

Solve for $x$ .

Tap for more steps...

Step 1.2.1

Subtract $1$ from both sides of the equation.

$3 x = - 1$

Step 1.2.2

Divide each term in $3 x = - 1$ by $3$ and simplify.

Tap for more steps...

Step 1.2.2.1

Divide each term in $3 x = - 1$ by $3$ .

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

Step 1.2.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.2.2.2.1.1

Cancel the common factor.

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

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 1.2.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.2.3.1

Move the negative in front of the fraction.

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

Step 1.3

The vertical asymptote occurs at $x = - \frac{1}{3}$ .

Vertical Asymptote: $x = - \frac{1}{3}$

Step 2

Find the point at $x = 1$ .

Tap for more steps...

Step 2.1

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

$f (1) = \ln (3 (1) + 1)$

Step 2.2

Simplify the result.

Tap for more steps...

Step 2.2.1

Multiply $3$ by $1$ .

$f (1) = \ln (3 + 1)$

Step 2.2.2

Add $3$ and $1$ .

$f (1) = \ln (4)$

Step 2.2.3

The final answer is $\ln (4)$ .

$\ln (4)$

Step 2.3

Convert $\ln (4)$ to decimal.

$y = 1.38629436$

Step 3

Find the point at $x = 2$ .

Tap for more steps...

Step 3.1

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

$f (2) = \ln (3 (2) + 1)$

Step 3.2

Simplify the result.

Tap for more steps...

Step 3.2.1

Multiply $3$ by $2$ .

$f (2) = \ln (6 + 1)$

Step 3.2.2

Add $6$ and $1$ .

$f (2) = \ln (7)$

Step 3.2.3

The final answer is $\ln (7)$ .

$\ln (7)$

Step 3.3

Convert $\ln (7)$ to decimal.

$y = 1.94591014$

Step 4

Find the point at $x = 3$ .

Tap for more steps...

Step 4.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = \ln (3 (3) + 1)$

Step 4.2

Simplify the result.

Tap for more steps...

Step 4.2.1

Multiply $3$ by $3$ .

$f (3) = \ln (9 + 1)$

Step 4.2.2

Add $9$ and $1$ .

$f (3) = \ln (10)$

Step 4.2.3

The final answer is $\ln (10)$ .

$\ln (10)$

Step 4.3

Convert $\ln (10)$ to decimal.

$y = 2.30258509$

Step 5

The log function can be graphed using the vertical asymptote at $x = - \frac{1}{3}$ and the points $(1, 1.38629436), (2, 1.94591014), (3, 2.30258509)$ .

Vertical Asymptote: $x = - \frac{1}{3}$

$\begin{array}{c} x & y \\ 1 & 1.386 \\ 2 & 1.946 \\ 3 & 2.303 \end{array}$

Step 6