Pre-Algebra Examples

$\ln (4 x) + 2 \ln (x) = 1.5$

Step 1

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

$\ln (4 x) + 2 \ln (x) = 1.5$

Step 2

Simplify the left side.

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Step 2.1

Simplify $\ln (4 x) + 2 \ln (x)$ .

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Step 2.1.1

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

$\ln (4 x) + \ln (x^{2}) = 1.5$

Step 2.1.2

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

$\ln (4 x \cdot x^{2}) = 1.5$

Step 2.1.3

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

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Step 2.1.3.1

Move $x^{2}$ .

$\ln (4 (x^{2} x)) = 1.5$

Step 2.1.3.2

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

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Step 2.1.3.2.1

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

$\ln (4 (x^{2} x^{1})) = 1.5$

Step 2.1.3.2.2

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

$\ln (4 x^{2 + 1}) = 1.5$

Step 2.1.3.3

Add $2$ and $1$ .

$\ln (4 x^{3}) = 1.5$

Step 3

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

$e^{\ln (4 x^{3})} = e^{1.5}$

Step 4

Rewrite $\ln (4 x^{3}) = 1.5$ 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 $b^{y} = x$ .

$e^{1.5} = 4 x^{3}$

Step 5

Solve for $x$ .

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Step 5.1

Rewrite the equation as $4 x^{3} = e^{1.5}$ .

$4 x^{3} = e^{1.5}$

Step 5.2

Divide each term in $4 x^{3} = e^{1.5}$ by $4$ and simplify.

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Step 5.2.1

Divide each term in $4 x^{3} = e^{1.5}$ by $4$ .

$\frac{4 x^{3}}{4} = \frac{e^{1.5}}{4}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $4$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{4 x^{3}}{4} = \frac{e^{1.5}}{4}$

Step 5.2.2.1.2

Divide $x^{3}$ by $1$ .

$x^{3} = \frac{e^{1.5}}{4}$

Step 5.3

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

$x = \sqrt[3]{\frac{e^{1.5}}{4}}$

Step 5.4

Simplify $\sqrt[3]{\frac{e^{1.5}}{4}}$ .

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Step 5.4.1

Rewrite $\sqrt[3]{\frac{e^{1.5}}{4}}$ as $\frac{\sqrt[3]{e^{1.5}}}{\sqrt[3]{4}}$ .

$x = \frac{\sqrt[3]{e^{1.5}}}{\sqrt[3]{4}}$

Step 5.4.2

Multiply $\frac{\sqrt[3]{e^{1.5}}}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$x = \frac{\sqrt[3]{e^{1.5}}}{\sqrt[3]{4}} \cdot \frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$

Step 5.4.3

Combine and simplify the denominator.

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Step 5.4.3.1

Multiply $\frac{\sqrt[3]{e^{1.5}}}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$x = \frac{\sqrt[3]{e^{1.5}} {\sqrt[3]{4}}^{2}}{\sqrt[3]{4} {\sqrt[3]{4}}^{2}}$

Step 5.4.3.2

Raise $\sqrt[3]{4}$ to the power of $1$ .

$x = \frac{\sqrt[3]{e^{1.5}} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1} {\sqrt[3]{4}}^{2}}$

Step 5.4.3.3

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

$x = \frac{\sqrt[3]{e^{1.5}} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1 + 2}}$

Step 5.4.3.4

Add $1$ and $2$ .

$x = \frac{\sqrt[3]{e^{1.5}} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{3}}$

Step 5.4.3.5

Rewrite ${\sqrt[3]{4}}^{3}$ as $4$ .

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Step 5.4.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4}$ as $4^{\frac{1}{3}}$ .

$x = \frac{\sqrt[3]{e^{1.5}} {\sqrt[3]{4}}^{2}}{{(4^{\frac{1}{3}})}^{3}}$

Step 5.4.3.5.2

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

$x = \frac{\sqrt[3]{e^{1.5}} {\sqrt[3]{4}}^{2}}{4^{\frac{1}{3} \cdot 3}}$

Step 5.4.3.5.3

Combine $\frac{1}{3}$ and $3$ .

$x = \frac{\sqrt[3]{e^{1.5}} {\sqrt[3]{4}}^{2}}{4^{\frac{3}{3}}}$

Step 5.4.3.5.4

Cancel the common factor of $3$ .

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Step 5.4.3.5.4.1

Cancel the common factor.

$x = \frac{\sqrt[3]{e^{1.5}} {\sqrt[3]{4}}^{2}}{4^{\frac{3}{3}}}$

Step 5.4.3.5.4.2

Rewrite the expression.

$x = \frac{\sqrt[3]{e^{1.5}} {\sqrt[3]{4}}^{2}}{4^{1}}$

Step 5.4.3.5.5

Evaluate the exponent.

$x = \frac{\sqrt[3]{e^{1.5}} {\sqrt[3]{4}}^{2}}{4}$

Step 5.4.4

Simplify the numerator.

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Step 5.4.4.1

Rewrite ${\sqrt[3]{4}}^{2}$ as $\sqrt[3]{4^{2}}$ .

$x = \frac{\sqrt[3]{e^{1.5}} \sqrt[3]{4^{2}}}{4}$

Step 5.4.4.2

Raise $4$ to the power of $2$ .

$x = \frac{\sqrt[3]{e^{1.5}} \sqrt[3]{16}}{4}$

Step 5.4.4.3

Rewrite $16$ as $2^{3} \cdot 2$ .

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Step 5.4.4.3.1

Factor $8$ out of $16$ .

$x = \frac{\sqrt[3]{e^{1.5}} \sqrt[3]{8 (2)}}{4}$

Step 5.4.4.3.2

Rewrite $8$ as $2^{3}$ .

$x = \frac{\sqrt[3]{e^{1.5}} \sqrt[3]{2^{3} \cdot 2}}{4}$

Step 5.4.4.4

Pull terms out from under the radical.

$x = \frac{\sqrt[3]{e^{1.5}} \cdot 2 \sqrt[3]{2}}{4}$

Step 5.4.4.5

Combine using the product rule for radicals.

$x = \frac{2 \sqrt[3]{e^{1.5} \cdot 2}}{4}$

Step 5.4.5

Reduce the expression by cancelling the common factors.

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Step 5.4.5.1

Cancel the common factor of $2$ and $4$ .

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Step 5.4.5.1.1

Factor $2$ out of $2 \sqrt[3]{e^{1.5} \cdot 2}$ .

$x = \frac{2 (\sqrt[3]{e^{1.5} \cdot 2})}{4}$

Step 5.4.5.1.2

Cancel the common factors.

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Step 5.4.5.1.2.1

Factor $2$ out of $4$ .

$x = \frac{2 \sqrt[3]{e^{1.5} \cdot 2}}{2 \cdot 2}$

Step 5.4.5.1.2.2

Cancel the common factor.

$x = \frac{2 \sqrt[3]{e^{1.5} \cdot 2}}{2 \cdot 2}$

Step 5.4.5.1.2.3

Rewrite the expression.

$x = \frac{\sqrt[3]{e^{1.5} \cdot 2}}{2}$

Step 5.4.5.2

Reorder factors in $\frac{\sqrt[3]{e^{1.5} \cdot 2}}{2}$ .

$x = \frac{\sqrt[3]{2 e^{1.5}}}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt[3]{2 e^{1.5}}}{2}$

Decimal Form:

$x = 1.03862931 \dots$