Algebra Examples

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

Step 1

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

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

Step 2

Simplify the left side.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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Step 2.1.3.1

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

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

Step 2.1.3.2

Add $2$ and $2$ .

$\ln (x^{4}) = 3$

Step 3

Write in exponential form.

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Step 3.1

For logarithmic equations, $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ such that $x > 0$ , $b > 0$ , and $b \neq 1$ . In this case, $b = e$ , $x = x^{4}$ , and $y = 3$ .

$b = e$

$x = x^{4}$

$y = 3$

Step 3.2

Substitute the values of $b$ , $x$ , and $y$ into the equation $b^{y} = x$ .

$e^{3} = x^{4}$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $x^{4} = e^{3}$ .

$x^{4} = e^{3}$

Step 4.2

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

$x = \pm \sqrt[4]{e^{3}}$

Step 4.3

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

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Step 4.3.1

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

$x = \sqrt[4]{e^{3}}$

Step 4.3.2

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

$x = - \sqrt[4]{e^{3}}$

Step 4.3.3

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

$x = \sqrt[4]{e^{3}}, - \sqrt[4]{e^{3}}$

Step 5

Exclude the solutions that do not make $2 \ln (x) + \ln (x^{2}) = 3$ true.

$x = \sqrt[4]{e^{3}}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt[4]{e^{3}}$

Decimal Form:

$x = 2.11700001 \dots$