Trigonometry Examples

$\ln (x) + \ln (x^{2})$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$x^{3} = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

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

$x = \sqrt[3]{0}$

Step 1.2.2

Simplify $\sqrt[3]{0}$ .

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Step 1.2.2.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 1.2.2.2

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

$x = 0$

Step 1.3

The vertical asymptote occurs at $x = 0$ .

Vertical Asymptote: $x = 0$

Step 2

Find the point at $x = 1$ .

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Step 2.1

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

$f (1) = \ln (1) + \ln ({(1)}^{2})$

Step 2.2

Simplify the result.

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Step 2.2.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$f (1) = \ln (1 {(1)}^{2})$

Step 2.2.2

Multiply ${(1)}^{2}$ by $1$ .

$f (1) = \ln ({(1)}^{2})$

Step 2.2.3

One to any power is one.

$f (1) = \ln (1)$

Step 2.2.4

The natural logarithm of $1$ is $0$ .

$f (1) = 0$

Step 2.2.5

The final answer is $0$ .

$0$

Step 2.3

Convert $0$ to decimal.

$y = 0$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

$f (2) = \ln (2) + \ln ({(2)}^{2})$

Step 3.2

Simplify the result.

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Step 3.2.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$f (2) = \ln (2 {(2)}^{2})$

Step 3.2.2

Multiply $2$ by ${(2)}^{2}$ by adding the exponents.

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Step 3.2.2.1

Multiply $2$ by ${(2)}^{2}$ .

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Step 3.2.2.1.1

Raise $2$ to the power of $1$ .

$f (2) = \ln (2 {(2)}^{2})$

Step 3.2.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (2) = \ln (2^{1 + 2})$

Step 3.2.2.2

Add $1$ and $2$ .

$f (2) = \ln (2^{3})$

Step 3.2.3

Raise $2$ to the power of $3$ .

$f (2) = \ln (8)$

Step 3.2.4

The final answer is $\ln (8)$ .

$\ln (8)$

Step 3.3

Convert $\ln (8)$ to decimal.

$y = 2.07944154$

Step 4

Find the point at $x = 3$ .

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Step 4.1

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

$f (3) = \ln (3) + \ln ({(3)}^{2})$

Step 4.2

Simplify the result.

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Step 4.2.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$f (3) = \ln (3 {(3)}^{2})$

Step 4.2.2

Multiply $3$ by ${(3)}^{2}$ by adding the exponents.

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Step 4.2.2.1

Multiply $3$ by ${(3)}^{2}$ .

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Step 4.2.2.1.1

Raise $3$ to the power of $1$ .

$f (3) = \ln (3 {(3)}^{2})$

Step 4.2.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (3) = \ln (3^{1 + 2})$

Step 4.2.2.2

Add $1$ and $2$ .

$f (3) = \ln (3^{3})$

Step 4.2.3

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

$f (3) = \ln (27)$

Step 4.2.4

The final answer is $\ln (27)$ .

$\ln (27)$

Step 4.3

Convert $\ln (27)$ to decimal.

$y = 3.29583686$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 0), (2, 2.07944154), (3, 3.29583686)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 0 \\ 2 & 2.079 \\ 3 & 3.296 \end{array}$

Step 6