Algebra Examples

$f (x) = \log_{\frac{1}{3}} (x + 1) - 2$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \log_{\frac{1}{3}} (x + 1) - 2$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\log_{\frac{1}{3}} (x + 1) - 2 = 0$ .

$\log_{\frac{1}{3}} (x + 1) - 2 = 0$

Step 1.2.2

Add $2$ to both sides of the equation.

$\log_{\frac{1}{3}} (x + 1) = 2$

Step 1.2.3

Rewrite $\log_{\frac{1}{3}} (x + 1) = 2$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

${(\frac{1}{3})}^{2} = x + 1$

Step 1.2.4

Solve for $x$ .

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Step 1.2.4.1

Rewrite the equation as $x + 1 = {(\frac{1}{3})}^{2}$ .

$x + 1 = {(\frac{1}{3})}^{2}$

Step 1.2.4.2

Simplify ${(\frac{1}{3})}^{2}$ .

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Step 1.2.4.2.1

Apply the product rule to $\frac{1}{3}$ .

$x + 1 = \frac{1^{2}}{3^{2}}$

Step 1.2.4.2.2

One to any power is one.

$x + 1 = \frac{1}{3^{2}}$

Step 1.2.4.2.3

Raise $3$ to the power of $2$ .

$x + 1 = \frac{1}{9}$

Step 1.2.4.3

Move all terms not containing $x$ to the right side of the equation.

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Step 1.2.4.3.1

Subtract $1$ from both sides of the equation.

$x = \frac{1}{9} - 1$

Step 1.2.4.3.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$x = \frac{1}{9} - 1 \cdot \frac{9}{9}$

Step 1.2.4.3.3

Combine $- 1$ and $\frac{9}{9}$ .

$x = \frac{1}{9} + \frac{- 1 \cdot 9}{9}$

Step 1.2.4.3.4

Combine the numerators over the common denominator.

$x = \frac{1 - 1 \cdot 9}{9}$

Step 1.2.4.3.5

Simplify the numerator.

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Step 1.2.4.3.5.1

Multiply $- 1$ by $9$ .

$x = \frac{1 - 9}{9}$

Step 1.2.4.3.5.2

Subtract $9$ from $1$ .

$x = \frac{- 8}{9}$

Step 1.2.4.3.6

Move the negative in front of the fraction.

$x = - \frac{8}{9}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- \frac{8}{9}, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \log_{\frac{1}{3}} ((0) + 1) - 2$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \log_{\frac{1}{3}} (0 + 1) - 2$

Step 2.2.2

Remove parentheses.

$y = \log_{\frac{1}{3}} ((0) + 1) - 2$

Step 2.2.3

Simplify $\log_{\frac{1}{3}} ((0) + 1) - 2$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Add $0$ and $1$ .

$y = \log_{\frac{1}{3}} (1) - 2$

Step 2.2.3.1.2

Logarithm base $\frac{1}{3}$ of $1$ is $0$ .

$y = 0 - 2$

Step 2.2.3.2

Subtract $2$ from $0$ .

$y = - 2$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 2)$

Step 3

List the intersections.

x-intercept(s): $(- \frac{8}{9}, 0)$

y-intercept(s): $(0, - 2)$

Step 4