Algebra Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

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

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

Step 1.2.2

Multiply both sides of the equation by $- 2$ .

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

Step 1.2.3

Simplify both sides of the equation.

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Step 1.2.3.1

Simplify the left side.

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Step 1.2.3.1.1

Simplify $- 2 (- \frac{1}{2} \cdot \ln (x + 3))$ .

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Step 1.2.3.1.1.1

Combine $\ln (x + 3)$ and $\frac{1}{2}$ .

$- 2 (- \frac{\ln (x + 3)}{2}) = - 2 \cdot 0$

Step 1.2.3.1.1.2

Cancel the common factor of $2$ .

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Step 1.2.3.1.1.2.1

Move the leading negative in $- \frac{\ln (x + 3)}{2}$ into the numerator.

$- 2 \frac{- \ln (x + 3)}{2} = - 2 \cdot 0$

Step 1.2.3.1.1.2.2

Factor $2$ out of $- 2$ .

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

Step 1.2.3.1.1.2.3

Cancel the common factor.

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

Step 1.2.3.1.1.2.4

Rewrite the expression.

$- - \ln (x + 3) = - 2 \cdot 0$

Step 1.2.3.1.1.3

Multiply.

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Step 1.2.3.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 \ln (x + 3) = - 2 \cdot 0$

Step 1.2.3.1.1.3.2

Multiply $\ln (x + 3)$ by $1$ .

$\ln (x + 3) = - 2 \cdot 0$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply $- 2$ by $0$ .

$\ln (x + 3) = 0$

Step 1.2.4

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (x + 3)} = e^{0}$

Step 1.2.5

Rewrite $\ln (x + 3) = 0$ 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 $b^{y} = x$ .

$e^{0} = x + 3$

Step 1.2.6

Solve for $x$ .

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Step 1.2.6.1

Rewrite the equation as $x + 3 = e^{0}$ .

$x + 3 = e^{0}$

Step 1.2.6.2

Anything raised to $0$ is $1$ .

$x + 3 = 1$

Step 1.2.6.3

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

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Step 1.2.6.3.1

Subtract $3$ from both sides of the equation.

$x = 1 - 3$

Step 1.2.6.3.2

Subtract $3$ from $1$ .

$x = - 2$

Step 1.3

x-intercept(s) in point form.

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

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

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

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Step 2.2.3.1

Add $0$ and $3$ .

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

Step 2.2.3.2

Multiply $- \frac{1}{2} \ln (3)$ .

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Step 2.2.3.2.1

Reorder $\ln (3)$ and $\frac{1}{2}$ .

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

Step 2.2.3.2.2

Simplify $\frac{1}{2} \ln (3)$ by moving $\frac{1}{2}$ inside the logarithm.

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

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \ln (3^{\frac{1}{2}}))$

Step 3

List the intersections.

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

y-intercept(s): $(0, - \ln (3^{\frac{1}{2}}))$

Step 4