Pre-Algebra Examples

$\log (x^{2} - 441) - 5 \log (x + 21)$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$\frac{x - 21}{{(x + 21)}^{4}} = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Set the numerator equal to zero.

$x - 21 = 0$

Step 1.2.2

Add $21$ to both sides of the equation.

$x = 21$

Step 1.3

The vertical asymptote occurs at $x = 21$ .

Vertical Asymptote: $x = 21$

Step 2

Find the point at $x = 22$ .

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Step 2.1

Replace the variable $x$ with $22$ in the expression.

$f (22) = \log ({(22)}^{2} - 441) - 5 \log ((22) + 21)$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Raise $22$ to the power of $2$ .

$f (22) = \log (484 - 441) - 5 \log ((22) + 21)$

Step 2.2.1.2

Subtract $441$ from $484$ .

$f (22) = \log (43) - 5 \log ((22) + 21)$

Step 2.2.1.3

Add $22$ and $21$ .

$f (22) = \log (43) - 5 \log (43)$

Step 2.2.2

Subtract $5 \log (43)$ from $\log (43)$ .

$f (22) = - 4 \log (43)$

Step 2.2.3

Simplify $- 4 \log (43)$ by moving $4$ inside the logarithm.

$f (22) = - \log (43^{4})$

Step 2.2.4

Raise $43$ to the power of $4$ .

$f (22) = - \log (3418801)$

Step 2.2.5

The final answer is $- \log (3418801)$ .

$- \log (3418801)$

Step 2.3

Convert $- \log (3418801)$ to decimal.

$y = - 6.53387382$

Step 3

Find the point at $x = 23$ .

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Step 3.1

Replace the variable $x$ with $23$ in the expression.

$f (23) = \log ({(23)}^{2} - 441) - 5 \log ((23) + 21)$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Raise $23$ to the power of $2$ .

$f (23) = \log (529 - 441) - 5 \log ((23) + 21)$

Step 3.2.1.2

Subtract $441$ from $529$ .

$f (23) = \log (88) - 5 \log ((23) + 21)$

Step 3.2.1.3

Add $23$ and $21$ .

$f (23) = \log (88) - 5 \log (44)$

Step 3.2.2

The final answer is $\log (88) - 5 \log (44)$ .

$\log (88) - 5 \log (44)$

Step 3.3

Convert $\log (88) - 5 \log (44)$ to decimal.

$y = - 6.27278071$

Step 4

Find the point at $x = 24$ .

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Step 4.1

Replace the variable $x$ with $24$ in the expression.

$f (24) = \log ({(24)}^{2} - 441) - 5 \log ((24) + 21)$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Raise $24$ to the power of $2$ .

$f (24) = \log (576 - 441) - 5 \log ((24) + 21)$

Step 4.2.1.2

Subtract $441$ from $576$ .

$f (24) = \log (135) - 5 \log ((24) + 21)$

Step 4.2.1.3

Add $24$ and $21$ .

$f (24) = \log (135) - 5 \log (45)$

Step 4.2.2

The final answer is $\log (135) - 5 \log (45)$ .

$\log (135) - 5 \log (45)$

Step 4.3

Convert $\log (135) - 5 \log (45)$ to decimal.

$y = - 6.1357288$

Step 5

The log function can be graphed using the vertical asymptote at $x = 21$ and the points $(22, - 6.53387382), (23, - 6.27278071), (24, - 6.1357288)$ .

Vertical Asymptote: $x = 21$

$\begin{array}{c} x & y \\ 22 & - 6.534 \\ 23 & - 6.273 \\ 24 & - 6.136 \end{array}$

Step 6