Trigonometry Examples

$\log_{3} (\frac{81}{\sqrt{x - 1}})$

Step 1

Find the domain for $y = \log_{3} (\frac{81}{\sqrt{x - 1}})$ so that a list of $x$ values can be picked to find a list of points, which will help graphing the radical.

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Step 1.1

Set the argument in $\log_{3} (\frac{81 \sqrt{x - 1}}{x - 1})$ greater than $0$ to find where the expression is defined.

$\frac{81 \sqrt{x - 1}}{x - 1} > 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Cross multiply.

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Step 1.2.1.1

Cross multiply by setting the product of the numerator of the right side and the denominator of the left side equal to the product of the numerator of the left side and the denominator of the right side.

$0 \cdot (x - 1) > 81 \sqrt{x - 1}$

Step 1.2.1.2

Simplify the left side.

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Step 1.2.1.2.1

Multiply $0$ by $x - 1$ .

$0 > 81 \sqrt{x - 1}$

Step 1.2.2

Rewrite so $81 \sqrt{x - 1}$ is on the left side of the inequality.

$81 \sqrt{x - 1} < 0$

Step 1.2.3

To remove the radical on the left side of the inequality, square both sides of the inequality.

${(81 \sqrt{x - 1})}^{2} < 0^{2}$

Step 1.2.4

Simplify each side of the inequality.

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Step 1.2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 1}$ as ${(x - 1)}^{\frac{1}{2}}$ .

${(81 {(x - 1)}^{\frac{1}{2}})}^{2} < 0^{2}$

Step 1.2.4.2

Simplify the left side.

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Step 1.2.4.2.1

Simplify ${(81 {(x - 1)}^{\frac{1}{2}})}^{2}$ .

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Step 1.2.4.2.1.1

Apply the product rule to $81 {(x - 1)}^{\frac{1}{2}}$ .

$81^{2} {({(x - 1)}^{\frac{1}{2}})}^{2} < 0^{2}$

Step 1.2.4.2.1.2

Raise $81$ to the power of $2$ .

$6561 {({(x - 1)}^{\frac{1}{2}})}^{2} < 0^{2}$

Step 1.2.4.2.1.3

Multiply the exponents in ${({(x - 1)}^{\frac{1}{2}})}^{2}$ .

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Step 1.2.4.2.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$6561 {(x - 1)}^{\frac{1}{2} \cdot 2} < 0^{2}$

Step 1.2.4.2.1.3.2

Cancel the common factor of $2$ .

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Step 1.2.4.2.1.3.2.1

Cancel the common factor.

$6561 {(x - 1)}^{\frac{1}{2} \cdot 2} < 0^{2}$

Step 1.2.4.2.1.3.2.2

Rewrite the expression.

$6561 {(x - 1)}^{1} < 0^{2}$

Step 1.2.4.2.1.4

Simplify.

$6561 (x - 1) < 0^{2}$

Step 1.2.4.2.1.5

Apply the distributive property.

$6561 x + 6561 \cdot - 1 < 0^{2}$

Step 1.2.4.2.1.6

Multiply $6561$ by $- 1$ .

$6561 x - 6561 < 0^{2}$

Step 1.2.4.3

Simplify the right side.

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Step 1.2.4.3.1

Raising $0$ to any positive power yields $0$ .

$6561 x - 6561 < 0$

Step 1.2.5

Solve for $x$ .

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Step 1.2.5.1

Add $6561$ to both sides of the inequality.

$6561 x < 6561$

Step 1.2.5.2

Divide each term in $6561 x < 6561$ by $6561$ and simplify.

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Step 1.2.5.2.1

Divide each term in $6561 x < 6561$ by $6561$ .

$\frac{6561 x}{6561} < \frac{6561}{6561}$

Step 1.2.5.2.2

Simplify the left side.

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Step 1.2.5.2.2.1

Cancel the common factor of $6561$ .

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Step 1.2.5.2.2.1.1

Cancel the common factor.

$\frac{6561 x}{6561} < \frac{6561}{6561}$

Step 1.2.5.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{6561}{6561}$

Step 1.2.5.2.3

Simplify the right side.

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Step 1.2.5.2.3.1

Divide $6561$ by $6561$ .

$x < 1$

Step 1.2.6

Find the domain of $\frac{81 \sqrt{x - 1}}{x - 1}$ .

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Step 1.2.6.1

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

$x - 1 \geq 0$

Step 1.2.6.2

Add $1$ to both sides of the inequality.

$x \geq 1$

Step 1.2.6.3

Set the denominator in $\frac{81 \sqrt{x - 1}}{x - 1}$ equal to $0$ to find where the expression is undefined.

$x - 1 = 0$

Step 1.2.6.4

Add $1$ to both sides of the equation.

$x = 1$

Step 1.2.6.5

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

$(1, \infty)$

Step 1.2.7

The solution consists of all of the true intervals.

$x > 1$

Step 1.3

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

$x - 1 \geq 0$

Step 1.4

Add $1$ to both sides of the inequality.

$x \geq 1$

Step 1.5

Set the denominator in $\frac{81 \sqrt{x - 1}}{x - 1}$ equal to $0$ to find where the expression is undefined.

$x - 1 = 0$

Step 1.6

Add $1$ to both sides of the equation.

$x = 1$

Step 1.7

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

Interval Notation:

$(1, \infty)$

Set-Builder Notation:

${x | x > 1}$

Interval Notation:

$(1, \infty)$

Set-Builder Notation:

${x | x > 1}$

Step 2

To find the radical expression end point, substitute the $x$ value $1$ , which is the least value in the domain, into $f (x) = \log_{3} (\frac{81 \sqrt{x - 1}}{x - 1})$ .

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Step 2.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = \log_{3} (\frac{81 \sqrt{(1) - 1}}{(1) - 1})$

Step 2.2

Subtract $1$ from $1$ .

$f (1) = \log_{3} (\frac{81 \sqrt{(1) - 1}}{0})$

Step 2.3

The expression contains a division by $0$ . The expression is undefined.

$f (1) = Undefined$

Undefined

Step 3

The radical expression end point is $(1, Undefined)$ .

$(1, Undefined)$

Step 4

Select a few $x$ values from the domain. It would be more useful to select the values so that they are next to the $x$ value of the radical expression end point.

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Step 4.1

Substitute the $x$ value $2$ into $f (x) = \log_{3} (\frac{81 \sqrt{x - 1}}{x - 1})$ . In this case, the point is $(2, 4)$ .

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Step 4.1.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = \log_{3} (\frac{81 \sqrt{(2) - 1}}{(2) - 1})$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Simplify the numerator.

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Step 4.1.2.1.1

Subtract $1$ from $2$ .

$f (2) = \log_{3} (\frac{81 \sqrt{1}}{2 - 1})$

Step 4.1.2.1.2

Any root of $1$ is $1$ .

$f (2) = \log_{3} (\frac{81 \cdot 1}{2 - 1})$

Step 4.1.2.2

Simplify the expression.

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Step 4.1.2.2.1

Subtract $1$ from $2$ .

$f (2) = \log_{3} (\frac{81 \cdot 1}{1})$

Step 4.1.2.2.2

Multiply $81$ by $1$ .

$f (2) = \log_{3} (\frac{81}{1})$

Step 4.1.2.3

Logarithm base $3$ of $\frac{81}{1}$ is $4$ .

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Step 4.1.2.3.1

Rewrite as an equation.

$\log_{3} (\frac{81}{1}) = x$

Step 4.1.2.3.2

Rewrite $\log_{3} (\frac{81}{1}) = x$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b$ does not equal $1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$3^{x} = \frac{81}{1}$

Step 4.1.2.3.3

Create equivalent expressions in the equation that all have equal bases.

$3^{x} = 3^{4}$

Step 4.1.2.3.4

Since the bases are the same, the two expressions are only equal if the exponents are also equal.

$x = 4$

Step 4.1.2.3.5

The variable $x$ is equal to $4$ .

$f (2) = 4$

Step 4.1.2.4

The final answer is $4$ .

$y = 4$

Step 4.2

Substitute the $x$ value $3$ into $f (x) = \log_{3} (\frac{81 \sqrt{x - 1}}{x - 1})$ . In this case, the point is $(3, \log_{3} (\frac{81 \sqrt{2}}{2}))$ .

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Step 4.2.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = \log_{3} (\frac{81 \sqrt{(3) - 1}}{(3) - 1})$

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Subtract $1$ from $3$ .

$f (3) = \log_{3} (\frac{81 \sqrt{2}}{3 - 1})$

Step 4.2.2.2

Subtract $1$ from $3$ .

$f (3) = \log_{3} (\frac{81 \sqrt{2}}{2})$

Step 4.2.2.3

The final answer is $\log_{3} (\frac{81 \sqrt{2}}{2})$ .

$y = \log_{3} (\frac{81 \sqrt{2}}{2})$

Step 4.3

The square root can be graphed using the points around the vertex $(1, Undefined), (2, 4), (3, 3.68)$

$\begin{array}{c} x & y \\ 1 & Undefined \\ 2 & 4 \\ 3 & 3.68 \end{array}$

Step 5