Pre-Algebra Examples

$\log (5 - x) - \frac{1}{3} \cdot \log (35 - x^{3})$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$\frac{5 - x}{{(35 - x^{3})}^{\frac{1}{3}}} = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Set the numerator equal to zero.

$5 - x = 0$

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

Subtract $5$ from both sides of the equation.

$- x = - 5$

Step 1.2.2.2

Divide each term in $- x = - 5$ by $- 1$ and simplify.

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Step 1.2.2.2.1

Divide each term in $- x = - 5$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 5}{- 1}$

Step 1.2.2.2.2

Simplify the left side.

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Step 1.2.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 5}{- 1}$

Step 1.2.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 5}{- 1}$

Step 1.2.2.2.3

Simplify the right side.

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Step 1.2.2.2.3.1

Divide $- 5$ by $- 1$ .

$x = 5$

Step 1.3

The vertical asymptote occurs at $x = 5$ .

Vertical Asymptote: $x = 5$

Step 2

Find the point at $x = 3$ .

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Step 2.1

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

$f (3) = \log (5 - (3)) - \frac{1}{3} \cdot \log (35 - {(3)}^{3})$

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

Multiply $- 1$ by $3$ .

$f (3) = \log (5 - 3) - \frac{1}{3} \cdot \log (35 - {(3)}^{3})$

Step 2.2.1.2

Subtract $3$ from $5$ .

$f (3) = \log (2) - \frac{1}{3} \cdot \log (35 - {(3)}^{3})$

Step 2.2.1.3

Simplify each term.

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Step 2.2.1.3.1

Raise $3$ to the power of $3$ .

$f (3) = \log (2) - \frac{1}{3} \cdot \log (35 - 1 \cdot 27)$

Step 2.2.1.3.2

Multiply $- 1$ by $27$ .

$f (3) = \log (2) - \frac{1}{3} \cdot \log (35 - 27)$

Step 2.2.1.4

Subtract $27$ from $35$ .

$f (3) = \log (2) - \frac{1}{3} \cdot \log (8)$

Step 2.2.1.5

Multiply $- \frac{1}{3} \log (8)$ .

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Step 2.2.1.5.1

Reorder $\log (8)$ and $\frac{1}{3}$ .

$f (3) = \log (2) - (\frac{1}{3} \cdot \log (8))$

Step 2.2.1.5.2

Simplify $\frac{1}{3} \log (8)$ by moving $\frac{1}{3}$ inside the logarithm.

$f (3) = \log (2) - \log (8^{\frac{1}{3}})$

Step 2.2.1.6

Rewrite $8$ as $2^{3}$ .

$f (3) = \log (2) - \log ({(2^{3})}^{\frac{1}{3}})$

Step 2.2.1.7

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

$f (3) = \log (2) - \log (2^{3 (\frac{1}{3})})$

Step 2.2.1.8

Cancel the common factor of $3$ .

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Step 2.2.1.8.1

Cancel the common factor.

$f (3) = \log (2) - \log (2^{3 (\frac{1}{3})})$

Step 2.2.1.8.2

Rewrite the expression.

$f (3) = \log (2) - \log (2)$

Step 2.2.1.9

Evaluate the exponent.

$f (3) = \log (2) - \log (2)$

Step 2.2.2

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

$f (3) = \log (\frac{2}{2})$

Step 2.2.3

Divide $2$ by $2$ .

$f (3) = \log (1)$

Step 2.2.4

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

$f (3) = 0$

Step 2.2.5

The final answer is $0$ .

$0$

Step 2.3

Convert $0$ to decimal.

$y = 0$

Step 3

The log function can be graphed using the vertical asymptote at $x = 5$ and the points $(3, 0)$ .

Vertical Asymptote: $x = 5$

$\begin{array}{c} x & y \\ 3 & 0 \end{array}$

Step 4