Algebra Examples

$y = \frac{1}{2} \log (x + 1) - \log (2 x + 10)$

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 $x$ .

$0 = \frac{1}{2} \cdot \log (x + 1) - \log (2 x + 10)$

Step 1.2

Solve the equation.

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Step 1.2.1

Remove parentheses.

$0 = \frac{1}{2} \cdot \log (x + 1) - \log (2 x + 10)$

Step 1.2.2

Simplify $\frac{1}{2} \cdot \log (x + 1) - \log (2 x + 10)$ .

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Step 1.2.2.1

Simplify $\frac{1}{2} \log (x + 1)$ by moving $\frac{1}{2}$ inside the logarithm.

$0 = \log ({(x + 1)}^{\frac{1}{2}}) - \log (2 x + 10)$

Step 1.2.2.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$0 = \log (\frac{{(x + 1)}^{\frac{1}{2}}}{2 x + 10})$

Step 1.2.2.3

Factor $2$ out of $2 x + 10$ .

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Step 1.2.2.3.1

Factor $2$ out of $2 x$ .

$0 = \log (\frac{{(x + 1)}^{\frac{1}{2}}}{2 (x) + 10})$

Step 1.2.2.3.2

Factor $2$ out of $10$ .

$0 = \log (\frac{{(x + 1)}^{\frac{1}{2}}}{2 x + 2 \cdot 5})$

Step 1.2.2.3.3

Factor $2$ out of $2 x + 2 \cdot 5$ .

$0 = \log (\frac{{(x + 1)}^{\frac{1}{2}}}{2 (x + 5)})$

Step 1.2.3

Graph each side of the equation. The solution is the x-value of the point of intersection.

No solution

Step 1.3

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

x-intercept(s): $None$

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 $y$ .

$y = \frac{1}{2} \cdot \log ((0) + 1) - \log (2 (0) + 10)$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{1}{2} \cdot \log (0 + 1) - \log (2 (0) + 10)$

Step 2.2.2

Remove parentheses.

$y = \frac{1}{2} \cdot \log ((0) + 1) - \log (2 (0) + 10)$

Step 2.2.3

Simplify $\frac{1}{2} \cdot \log ((0) + 1) - \log (2 (0) + 10)$ .

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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 $1$ .

$y = \frac{1}{2} \cdot \log (1) - \log (2 (0) + 10)$

Step 2.2.3.1.2

Logarithm base $10$ of $1$ is $0$ .

$y = \frac{1}{2} \cdot 0 - \log (2 (0) + 10)$

Step 2.2.3.1.3

Multiply $\frac{1}{2}$ by $0$ .

$y = 0 - \log (2 (0) + 10)$

Step 2.2.3.1.4

Multiply $2$ by $0$ .

$y = 0 - \log (0 + 10)$

Step 2.2.3.1.5

Add $0$ and $10$ .

$y = 0 - \log (10)$

Step 2.2.3.1.6

Logarithm base $10$ of $10$ is $1$ .

$y = 0 - 1 \cdot 1$

Step 2.2.3.1.7

Multiply $- 1$ by $1$ .

$y = 0 - 1$

Step 2.2.3.2

Subtract $1$ from $0$ .

$y = - 1$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 1)$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, - 1)$

Step 4