Trigonometry Examples

$20 + \ln (5) = 2 \ln (x)$

Step 1

Rewrite the equation as $2 \ln (x) = 20 + \ln (5)$ .

$2 \ln (x) = 20 + \ln (5)$

Step 2

Move all the terms containing a logarithm to the left side of the equation.

$2 \ln (x) - \ln (5) = 20$

Step 3

Simplify the left side.

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Step 3.1

Simplify $2 \ln (x) - \ln (5)$ .

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Step 3.1.1

Simplify $2 \ln (x)$ by moving $2$ inside the logarithm.

$\ln (x^{2}) - \ln (5) = 20$

Step 3.1.2

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

$\ln (\frac{x^{2}}{5}) = 20$

Step 4

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (\frac{x^{2}}{5})} = e^{20}$

Step 5

Rewrite $\ln (\frac{x^{2}}{5}) = 20$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{20} = \frac{x^{2}}{5}$

Step 6

Solve for $x$ .

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Step 6.1

Rewrite the equation as $\frac{x^{2}}{5} = e^{20}$ .

$\frac{x^{2}}{5} = e^{20}$

Step 6.2

Multiply both sides of the equation by $5$ .

$5 \frac{x^{2}}{5} = 5 e^{20}$

Step 6.3

Simplify the left side.

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Step 6.3.1

Cancel the common factor of $5$ .

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Step 6.3.1.1

Cancel the common factor.

$5 \frac{x^{2}}{5} = 5 e^{20}$

Step 6.3.1.2

Rewrite the expression.

$x^{2} = 5 e^{20}$

Step 6.4

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

$x = \pm \sqrt{5 e^{20}}$

Step 6.5

Simplify $\pm \sqrt{5 e^{20}}$ .

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Step 6.5.1

Rewrite $5 e^{20}$ as ${({(e^{5})}^{2})}^{2} \cdot 5$ .

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Step 6.5.1.1

Rewrite $e^{20}$ as ${(e^{10})}^{2}$ .

$x = \pm \sqrt{5 {(e^{10})}^{2}}$

Step 6.5.1.2

Reorder $5$ and ${(e^{10})}^{2}$ .

$x = \pm \sqrt{{(e^{10})}^{2} \cdot 5}$

Step 6.5.1.3

Rewrite $e^{10}$ as ${(e^{5})}^{2}$ .

$x = \pm \sqrt{{({(e^{5})}^{2})}^{2} \cdot 5}$

Step 6.5.2

Pull terms out from under the radical.

$x = \pm {(e^{5})}^{2} \sqrt{5}$

Step 6.5.3

Multiply the exponents in ${(e^{5})}^{2}$ .

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Step 6.5.3.1

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

$x = \pm e^{5 \cdot 2} \sqrt{5}$

Step 6.5.3.2

Multiply $5$ by $2$ .

$x = \pm e^{10} \sqrt{5}$

Step 6.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 6.6.1

First, use the positive value of the $\pm$ to find the first solution.

$x = e^{10} \sqrt{5}$

Step 6.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - e^{10} \sqrt{5}$

Step 6.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = e^{10} \sqrt{5}, - e^{10} \sqrt{5}$

Step 7

Exclude the solutions that do not make $20 + \ln (5) = 2 \ln (x)$ true.

$x = e^{10} \sqrt{5}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = e^{10} \sqrt{5}$

Decimal Form:

$x = 49252.67482126 \dots$