Trigonometry Examples

$f (x) = \log (4 x^{2})$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$4 x^{2} = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Divide each term in $4 x^{2} = 0$ by $4$ and simplify.

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Step 1.2.1.1

Divide each term in $4 x^{2} = 0$ by $4$ .

$\frac{4 x^{2}}{4} = \frac{0}{4}$

Step 1.2.1.2

Simplify the left side.

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Step 1.2.1.2.1

Cancel the common factor of $4$ .

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Step 1.2.1.2.1.1

Cancel the common factor.

$\frac{4 x^{2}}{4} = \frac{0}{4}$

Step 1.2.1.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{0}{4}$

Step 1.2.1.3

Simplify the right side.

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Step 1.2.1.3.1

Divide $0$ by $4$ .

$x^{2} = 0$

Step 1.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 1.2.3

Simplify $\pm \sqrt{0}$ .

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Step 1.2.3.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 1.2.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 1.2.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 1.3

The vertical asymptote occurs at $x = 0$ .

Vertical Asymptote: $x = 0$

Step 2

Find the point at $x = 5$ .

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Step 2.1

Replace the variable $x$ with $5$ in the expression.

$f (5) = \log (4 {(5)}^{2})$

Step 2.2

Simplify the result.

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Step 2.2.1

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

$f (5) = \log (4 \cdot 25)$

Step 2.2.2

Multiply $4$ by $25$ .

$f (5) = \log (100)$

Step 2.2.3

Logarithm base $10$ of $100$ is $2$ .

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Step 2.2.3.1

Rewrite as an equation.

$\log (100) = x$

Step 2.2.3.2

Rewrite $\log (100) = x$ 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 $b^{y} = x$ .

$10^{x} = 100$

Step 2.2.3.3

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

$10^{x} = 10^{2}$

Step 2.2.3.4

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

$x = 2$

Step 2.2.3.5

The variable $x$ is equal to $2$ .

$f (5) = 2$

Step 2.2.4

The final answer is $2$ .

$2$

Step 2.3

Convert $2$ to decimal.

$y = 2$

Step 3

Find the point at $x = 1$ .

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Step 3.1

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

$f (1) = \log (4 {(1)}^{2})$

Step 3.2

Simplify the result.

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Step 3.2.1

One to any power is one.

$f (1) = \log (4 \cdot 1)$

Step 3.2.2

Multiply $4$ by $1$ .

$f (1) = \log (4)$

Step 3.2.3

The final answer is $\log (4)$ .

$\log (4)$

Step 3.3

Convert $\log (4)$ to decimal.

$y = 0.60205999$

Step 4

Find the point at $x = 2$ .

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Step 4.1

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

$f (2) = \log (4 {(2)}^{2})$

Step 4.2

Simplify the result.

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Step 4.2.1

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

$f (2) = \log (4 \cdot 4)$

Step 4.2.2

Multiply $4$ by $4$ .

$f (2) = \log (16)$

Step 4.2.3

The final answer is $\log (16)$ .

$\log (16)$

Step 4.3

Convert $\log (16)$ to decimal.

$y = 1.20411998$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(5, 2), (1, 0.60205999), (2, 1.20411998)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 0.602 \\ 2 & 1.204 \\ 5 & 2 \end{array}$

Step 6