Precalculus Examples

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

Step 1

Subtract $3$ from both sides of the equation.

$\ln (x) + \ln (x^{2}) - 3 = 0$

Step 2

Simplify $\ln (x) + \ln (x^{2}) - 3$ .

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Step 2.1

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

$\ln (x \cdot x^{2}) - 3 = 0$

Step 2.2

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

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Step 2.2.1

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

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Step 2.2.1.1

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

$\ln (x^{1} x^{2}) - 3 = 0$

Step 2.2.1.2

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

$\ln (x^{1 + 2}) - 3 = 0$

Step 2.2.2

Add $1$ and $2$ .

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

Step 3

Add $3$ to both sides of the equation.

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

Step 4

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

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

Step 5

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

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

Step 6

Solve for $x$ .

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Step 6.1

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

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

Step 6.2

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$x = e$