Algebra Examples

$\ln (3 x) + \ln (2 x) = 3$

Step 1

Simplify the left side.

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Step 1.1

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

$\ln (3 x (2 x)) = 3$

Step 1.2

Rewrite using the commutative property of multiplication.

$\ln (3 \cdot (2 x \cdot x)) = 3$

Step 1.3

Multiply $x$ by $x$ by adding the exponents.

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Step 1.3.1

Move $x$ .

$\ln (3 \cdot (2 (x \cdot x))) = 3$

Step 1.3.2

Multiply $x$ by $x$ .

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

Step 1.4

Multiply $3$ by $2$ .

$\ln (6 x^{2}) = 3$

Step 2

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

$e^{\ln (6 x^{2})} = e^{3}$

Step 3

Rewrite $\ln (6 x^{2}) = 3$ 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 $b^{y} = x$ .

$e^{3} = 6 x^{2}$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $6 x^{2} = e^{3}$ .

$6 x^{2} = e^{3}$

Step 4.2

Divide each term in $6 x^{2} = e^{3}$ by $6$ and simplify.

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Step 4.2.1

Divide each term in $6 x^{2} = e^{3}$ by $6$ .

$\frac{6 x^{2}}{6} = \frac{e^{3}}{6}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $6$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{6 x^{2}}{6} = \frac{e^{3}}{6}$

Step 4.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{e^{3}}{6}$

Step 4.3

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

$x = \pm \sqrt{\frac{e^{3}}{6}}$

Step 4.4

Simplify $\pm \sqrt{\frac{e^{3}}{6}}$ .

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Step 4.4.1

Rewrite $\sqrt{\frac{e^{3}}{6}}$ as $\frac{\sqrt{e^{3}}}{\sqrt{6}}$ .

$x = \pm \frac{\sqrt{e^{3}}}{\sqrt{6}}$

Step 4.4.2

Simplify the numerator.

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Step 4.4.2.1

Factor out $e^{2}$ .

$x = \pm \frac{\sqrt{e^{2} e}}{\sqrt{6}}$

Step 4.4.2.2

Pull terms out from under the radical.

$x = \pm \frac{e \sqrt{e}}{\sqrt{6}}$

Step 4.4.3

Multiply $\frac{e \sqrt{e}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$x = \pm \frac{e \sqrt{e}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Step 4.4.4

Combine and simplify the denominator.

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Step 4.4.4.1

Multiply $\frac{e \sqrt{e}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$x = \pm \frac{e \sqrt{e} \sqrt{6}}{\sqrt{6} \sqrt{6}}$

Step 4.4.4.2

Raise $\sqrt{6}$ to the power of $1$ .

$x = \pm \frac{e \sqrt{e} \sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}}$

Step 4.4.4.3

Raise $\sqrt{6}$ to the power of $1$ .

$x = \pm \frac{e \sqrt{e} \sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}}$

Step 4.4.4.4

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

$x = \pm \frac{e \sqrt{e} \sqrt{6}}{{\sqrt{6}}^{1 + 1}}$

Step 4.4.4.5

Add $1$ and $1$ .

$x = \pm \frac{e \sqrt{e} \sqrt{6}}{{\sqrt{6}}^{2}}$

Step 4.4.4.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 4.4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$x = \pm \frac{e \sqrt{e} \sqrt{6}}{{(6^{\frac{1}{2}})}^{2}}$

Step 4.4.4.6.2

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

$x = \pm \frac{e \sqrt{e} \sqrt{6}}{6^{\frac{1}{2} \cdot 2}}$

Step 4.4.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{e \sqrt{e} \sqrt{6}}{6^{\frac{2}{2}}}$

Step 4.4.4.6.4

Cancel the common factor of $2$ .

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Step 4.4.4.6.4.1

Cancel the common factor.

$x = \pm \frac{e \sqrt{e} \sqrt{6}}{6^{\frac{2}{2}}}$

Step 4.4.4.6.4.2

Rewrite the expression.

$x = \pm \frac{e \sqrt{e} \sqrt{6}}{6^{1}}$

Step 4.4.4.6.5

Evaluate the exponent.

$x = \pm \frac{e \sqrt{e} \sqrt{6}}{6}$

Step 4.4.5

Combine using the product rule for radicals.

$x = \pm \frac{e \sqrt{6 e}}{6}$

Step 4.5

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

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Step 4.5.1

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

$x = \frac{e \sqrt{6 e}}{6}$

Step 4.5.2

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

$x = - \frac{e \sqrt{6 e}}{6}$

Step 4.5.3

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

$x = \frac{e \sqrt{6 e}}{6}, - \frac{e \sqrt{6 e}}{6}$

Step 5

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

$x = \frac{e \sqrt{6 e}}{6}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{e \sqrt{6 e}}{6}$

Decimal Form:

$x = 1.82964190 \dots$