Finite Math Examples

$5 \ln (8 x) = 20 - 2 \ln (x)$

Step 1

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

$5 \ln (8 x) + 2 \ln (x) = 20$

Step 2

Simplify the left side.

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Step 2.1

Simplify $5 \ln (8 x) + 2 \ln (x)$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Simplify $5 \ln (8 x)$ by moving $5$ inside the logarithm.

$\ln ({(8 x)}^{5}) + 2 \ln (x) = 20$

Step 2.1.1.2

Apply the product rule to $8 x$ .

$\ln (8^{5} x^{5}) + 2 \ln (x) = 20$

Step 2.1.1.3

Raise $8$ to the power of $5$ .

$\ln (32768 x^{5}) + 2 \ln (x) = 20$

Step 2.1.1.4

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

$\ln (32768 x^{5}) + \ln (x^{2}) = 20$

Step 2.1.2

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

$\ln (32768 x^{5} x^{2}) = 20$

Step 2.1.3

Multiply $x^{5}$ by $x^{2}$ by adding the exponents.

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Step 2.1.3.1

Move $x^{2}$ .

$\ln (32768 (x^{2} x^{5})) = 20$

Step 2.1.3.2

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

$\ln (32768 x^{2 + 5}) = 20$

Step 2.1.3.3

Add $2$ and $5$ .

$\ln (32768 x^{7}) = 20$

Step 3

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

$e^{\ln (32768 x^{7})} = e^{20}$

Step 4

Rewrite $\ln (32768 x^{7}) = 20$ 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 $b^{y} = x$ .

$e^{20} = 32768 x^{7}$

Step 5

Solve for $x$ .

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Step 5.1

Rewrite the equation as $32768 x^{7} = e^{20}$ .

$32768 x^{7} = e^{20}$

Step 5.2

Divide each term in $32768 x^{7} = e^{20}$ by $32768$ and simplify.

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Step 5.2.1

Divide each term in $32768 x^{7} = e^{20}$ by $32768$ .

$\frac{32768 x^{7}}{32768} = \frac{e^{20}}{32768}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $32768$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{32768 x^{7}}{32768} = \frac{e^{20}}{32768}$

Step 5.2.2.1.2

Divide $x^{7}$ by $1$ .

$x^{7} = \frac{e^{20}}{32768}$

Step 5.3

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

$x = \sqrt[7]{\frac{e^{20}}{32768}}$

Step 5.4

Simplify $\sqrt[7]{\frac{e^{20}}{32768}}$ .

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Step 5.4.1

Rewrite $\sqrt[7]{\frac{e^{20}}{32768}}$ as $\frac{\sqrt[7]{e^{20}}}{\sqrt[7]{32768}}$ .

$x = \frac{\sqrt[7]{e^{20}}}{\sqrt[7]{32768}}$

Step 5.4.2

Simplify the numerator.

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Step 5.4.2.1

Rewrite $e^{20}$ as ${(e^{2})}^{7} e^{6}$ .

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Step 5.4.2.1.1

Factor out $e^{14}$ .

$x = \frac{\sqrt[7]{e^{14} e^{6}}}{\sqrt[7]{32768}}$

Step 5.4.2.1.2

Rewrite $e^{14}$ as ${(e^{2})}^{7}$ .

$x = \frac{\sqrt[7]{{(e^{2})}^{7} e^{6}}}{\sqrt[7]{32768}}$

Step 5.4.2.2

Pull terms out from under the radical.

$x = \frac{e^{2} \sqrt[7]{e^{6}}}{\sqrt[7]{32768}}$

Step 5.4.3

Simplify the denominator.

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Step 5.4.3.1

Rewrite $32768$ as $4^{7} \cdot 2$ .

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Step 5.4.3.1.1

Factor $16384$ out of $32768$ .

$x = \frac{e^{2} \sqrt[7]{e^{6}}}{\sqrt[7]{16384 (2)}}$

Step 5.4.3.1.2

Rewrite $16384$ as $4^{7}$ .

$x = \frac{e^{2} \sqrt[7]{e^{6}}}{\sqrt[7]{4^{7} \cdot 2}}$

Step 5.4.3.2

Pull terms out from under the radical.

$x = \frac{e^{2} \sqrt[7]{e^{6}}}{4 \sqrt[7]{2}}$

Step 5.4.4

Multiply $\frac{e^{2} \sqrt[7]{e^{6}}}{4 \sqrt[7]{2}}$ by $\frac{{\sqrt[7]{2}}^{6}}{{\sqrt[7]{2}}^{6}}$ .

$x = \frac{e^{2} \sqrt[7]{e^{6}}}{4 \sqrt[7]{2}} \cdot \frac{{\sqrt[7]{2}}^{6}}{{\sqrt[7]{2}}^{6}}$

Step 5.4.5

Combine and simplify the denominator.

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Step 5.4.5.1

Multiply $\frac{e^{2} \sqrt[7]{e^{6}}}{4 \sqrt[7]{2}}$ by $\frac{{\sqrt[7]{2}}^{6}}{{\sqrt[7]{2}}^{6}}$ .

$x = \frac{e^{2} \sqrt[7]{e^{6}} {\sqrt[7]{2}}^{6}}{4 \sqrt[7]{2} {\sqrt[7]{2}}^{6}}$

Step 5.4.5.2

Move $\sqrt[7]{2}$ .

$x = \frac{e^{2} \sqrt[7]{e^{6}} {\sqrt[7]{2}}^{6}}{4 (\sqrt[7]{2} {\sqrt[7]{2}}^{6})}$

Step 5.4.5.3

Raise $\sqrt[7]{2}$ to the power of $1$ .

$x = \frac{e^{2} \sqrt[7]{e^{6}} {\sqrt[7]{2}}^{6}}{4 ({\sqrt[7]{2}}^{1} {\sqrt[7]{2}}^{6})}$

Step 5.4.5.4

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

$x = \frac{e^{2} \sqrt[7]{e^{6}} {\sqrt[7]{2}}^{6}}{4 {\sqrt[7]{2}}^{1 + 6}}$

Step 5.4.5.5

Add $1$ and $6$ .

$x = \frac{e^{2} \sqrt[7]{e^{6}} {\sqrt[7]{2}}^{6}}{4 {\sqrt[7]{2}}^{7}}$

Step 5.4.5.6

Rewrite ${\sqrt[7]{2}}^{7}$ as $2$ .

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Step 5.4.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{2}$ as $2^{\frac{1}{7}}$ .

$x = \frac{e^{2} \sqrt[7]{e^{6}} {\sqrt[7]{2}}^{6}}{4 {(2^{\frac{1}{7}})}^{7}}$

Step 5.4.5.6.2

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

$x = \frac{e^{2} \sqrt[7]{e^{6}} {\sqrt[7]{2}}^{6}}{4 \cdot 2^{\frac{1}{7} \cdot 7}}$

Step 5.4.5.6.3

Combine $\frac{1}{7}$ and $7$ .

$x = \frac{e^{2} \sqrt[7]{e^{6}} {\sqrt[7]{2}}^{6}}{4 \cdot 2^{\frac{7}{7}}}$

Step 5.4.5.6.4

Cancel the common factor of $7$ .

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Step 5.4.5.6.4.1

Cancel the common factor.

$x = \frac{e^{2} \sqrt[7]{e^{6}} {\sqrt[7]{2}}^{6}}{4 \cdot 2^{\frac{7}{7}}}$

Step 5.4.5.6.4.2

Rewrite the expression.

$x = \frac{e^{2} \sqrt[7]{e^{6}} {\sqrt[7]{2}}^{6}}{4 \cdot 2^{1}}$

Step 5.4.5.6.5

Evaluate the exponent.

$x = \frac{e^{2} \sqrt[7]{e^{6}} {\sqrt[7]{2}}^{6}}{4 \cdot 2}$

Step 5.4.6

Simplify the numerator.

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Step 5.4.6.1

Rewrite ${\sqrt[7]{2}}^{6}$ as $\sqrt[7]{2^{6}}$ .

$x = \frac{e^{2} \sqrt[7]{e^{6}} \sqrt[7]{2^{6}}}{4 \cdot 2}$

Step 5.4.6.2

Raise $2$ to the power of $6$ .

$x = \frac{e^{2} \sqrt[7]{e^{6}} \sqrt[7]{64}}{4 \cdot 2}$

Step 5.4.6.3

Combine using the product rule for radicals.

$x = \frac{e^{2} \sqrt[7]{64 e^{6}}}{4 \cdot 2}$

Step 5.4.7

Multiply $4$ by $2$ .

$x = \frac{e^{2} \sqrt[7]{64 e^{6}}}{8}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{e^{2} \sqrt[7]{64 e^{6}}}{8}$

Decimal Form:

$x = 3.94254900 \dots$