Precalculus Examples

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

Step 1

Simplify the left side.

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Step 1.1

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

$\ln (x \cdot x^{2}) = 3$

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 $1$ .

$\ln (x \cdot x^{2}) = 3$

Step 1.2.1.2

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

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

Step 1.2.2

Add $1$ and $2$ .

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

Step 2

Subtract $\ln (x^{3})$ from both sides of the equation.

$0 = 3 - \ln (x^{3})$

Step 3

Rewrite the equation as $3 - \ln (x^{3}) = 0$ .

$3 - \ln (x^{3}) = 0$