Finite Math Examples

$\sqrt{\log_{x} (x - 1)}$

Step 1

Set the base in $\log_{x} (x - 1)$ greater than $0$ to find where the expression is defined.

$x > 0$

Step 2

Set the argument in $\log_{x} (x - 1)$ greater than $0$ to find where the expression is defined.

$x - 1 > 0$

Step 3

Add $1$ to both sides of the inequality.

$x > 1$

Step 4

Set the radicand in $\sqrt{\log_{x} (x - 1)}$ greater than or equal to $0$ to find where the expression is defined.

$\log_{x} (x - 1) \geq 0$

Step 5

Solve for $x$ .

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Step 5.1

Convert the inequality to an equality.

$\log_{x} (x - 1) = 0$

Step 5.2

Solve the equation.

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Step 5.2.1

Rewrite $\log_{x} (x - 1) = 0$ 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 $b^{y} = x$ .

$x^{0} = x - 1$

Step 5.2.2

Solve for $x$ .

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Step 5.2.2.1

Anything raised to $0$ is $1$ .

$1 = x - 1$

Step 5.2.2.2

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x - 1 = 1$

Step 5.2.2.3

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

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Step 5.2.2.3.1

Add $1$ to both sides of the equation.

$x = 1 + 1$

Step 5.2.2.3.2

Add $1$ and $1$ .

$x = 2$

Step 5.3

Find the domain of $\log_{x} (x - 1)$ .

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Step 5.3.1

Set the base in $\log_{x} (x - 1)$ greater than $0$ to find where the expression is defined.

$x > 0$

Step 5.3.2

Set the argument in $\log_{x} (x - 1)$ greater than $0$ to find where the expression is defined.

$x - 1 > 0$

Step 5.3.3

Add $1$ to both sides of the inequality.

$x > 1$

Step 5.3.4

Set the base in $\log_{x} (x - 1)$ equal to $1$ to find where the expression is undefined.

$x = 1$

Step 5.3.5

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

$(1, \infty)$

Step 5.4

The solution consists of all of the true intervals.

$x \geq 2$

Step 6

Set the base in $\log_{x} (x - 1)$ equal to $1$ to find where the expression is undefined.

$x = 1$

Step 7

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

Interval Notation:

$[2, \infty)$

Set-Builder Notation:

${x | x \geq 2}$

Step 8