Algebra Examples

$\log_{5} (x + 5) \geq 2$

Step 1

Convert the inequality to an equality.

$\log_{5} (x + 5) = 2$

Step 2

Solve the equation.

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Step 2.1

Rewrite $\log_{5} (x + 5) = 2$ 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 $b^{y} = x$ .

$5^{2} = x + 5$

Step 2.2

Solve for $x$ .

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Step 2.2.1

Rewrite the equation as $x + 5 = 5^{2}$ .

$x + 5 = 5^{2}$

Step 2.2.2

Raise $5$ to the power of $2$ .

$x + 5 = 25$

Step 2.2.3

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

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Step 2.2.3.1

Subtract $5$ from both sides of the equation.

$x = 25 - 5$

Step 2.2.3.2

Subtract $5$ from $25$ .

$x = 20$

Step 3

Find the domain of $\log_{5} (x + 5) - 2$ .

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Step 3.1

Set the argument in $\log_{5} (x + 5)$ greater than $0$ to find where the expression is defined.

$x + 5 > 0$

Step 3.2

Subtract $5$ from both sides of the inequality.

$x > - 5$

Step 3.3

The domain is all values of $x$ that make the expression defined.

$(- 5, \infty)$

Step 4

The solution consists of all of the true intervals.

$x \geq 20$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x \geq 20$

Interval Notation:

$[20, \infty)$

Step 6