Precalculus Examples

$\ln (\sqrt{x + 7}) = 2$

Step 1

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

$e^{\ln (\sqrt{x + 7})} = e^{2}$

Step 2

Rewrite $\ln (\sqrt{x + 7}) = 2$ 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 $b^{y} = x$ .

$e^{2} = \sqrt{x + 7}$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $\sqrt{x + 7} = e^{2}$ .

$\sqrt{x + 7} = e^{2}$

Step 3.2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x + 7}}^{2} = {(e^{2})}^{2}$

Step 3.3

Simplify each side of the equation.

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Step 3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 7}$ as ${(x + 7)}^{\frac{1}{2}}$ .

${({(x + 7)}^{\frac{1}{2}})}^{2} = {(e^{2})}^{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Simplify ${({(x + 7)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.1

Multiply the exponents in ${({(x + 7)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.1.1

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

${(x + 7)}^{\frac{1}{2} \cdot 2} = {(e^{2})}^{2}$

Step 3.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.3.2.1.1.2.1

Cancel the common factor.

${(x + 7)}^{\frac{1}{2} \cdot 2} = {(e^{2})}^{2}$

Step 3.3.2.1.1.2.2

Rewrite the expression.

${(x + 7)}^{1} = {(e^{2})}^{2}$

Step 3.3.2.1.2

Simplify.

$x + 7 = {(e^{2})}^{2}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

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

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Step 3.3.3.1.1

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

$x + 7 = e^{2 \cdot 2}$

Step 3.3.3.1.2

Multiply $2$ by $2$ .

$x + 7 = e^{4}$

Step 3.4

Subtract $7$ from both sides of the equation.

$x = e^{4} - 7$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = e^{4} - 7$

Decimal Form:

$x = 47.59815003 \dots$