Precalculus Examples

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

Step 1

Simplify the left side.

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Step 1.1

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

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

Step 1.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 1.2.1

Multiply $x$ by $x^{2}$ .

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Step 1.2.1.1

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

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

Step 1.2.1.2

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

$\ln (x^{1 + 2}) = 5$

Step 1.2.2

Add $1$ and $2$ .

$\ln (x^{3}) = 5$

Step 2

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

$e^{\ln (x^{3})} = e^{5}$

Step 3

Rewrite $\ln (x^{3}) = 5$ 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 $b^{y} = x$ .

$e^{5} = x^{3}$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $x^{3} = e^{5}$ .

$x^{3} = e^{5}$

Step 4.2

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

$x = \sqrt[3]{e^{5}}$

Step 4.3

Simplify $\sqrt[3]{e^{5}}$ .

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Step 4.3.1

Factor out $e^{3}$ .

$x = \sqrt[3]{e^{3} e^{2}}$

Step 4.3.2

Pull terms out from under the radical.

$x = e \sqrt[3]{e^{2}}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = e \sqrt[3]{e^{2}}$

Decimal Form:

$x = 5.29449005 \dots$