Finite Math Examples

$\frac{1}{x^{4}} = {(\frac{1}{x})}^{y - 1}$

Step 1

Rewrite the equation as ${(\frac{1}{x})}^{y - 1} = \frac{1}{x^{4}}$ .

${(\frac{1}{x})}^{y - 1} = \frac{1}{x^{4}}$

Step 2

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln ({(\frac{1}{x})}^{y - 1}) = \ln (\frac{1}{x^{4}})$

Step 3

Expand the left side.

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Step 3.1

Expand $\ln ({(\frac{1}{x})}^{y - 1})$ by moving $y - 1$ outside the logarithm.

$(y - 1) \ln (\frac{1}{x}) = \ln (\frac{1}{x^{4}})$

Step 3.2

Rewrite $\ln (\frac{1}{x})$ as $\ln (1) - \ln (x)$ .

$(y - 1) (\ln (1) - \ln (x)) = \ln (\frac{1}{x^{4}})$

Step 3.3

The natural logarithm of $1$ is $0$ .

$(y - 1) (0 - \ln (x)) = \ln (\frac{1}{x^{4}})$

Step 3.4

Subtract $\ln (x)$ from $0$ .

$(y - 1) (- \ln (x)) = \ln (\frac{1}{x^{4}})$

Step 4

Simplify the left side.

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Step 4.1

Simplify $(y - 1) (- \ln (x))$ .

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Step 4.1.1

Apply the distributive property.

$y (- \ln (x)) - 1 (- \ln (x)) = \ln (\frac{1}{x^{4}})$

Step 4.1.2

Rewrite using the commutative property of multiplication.

$- y \ln (x) - 1 (- \ln (x)) = \ln (\frac{1}{x^{4}})$

Step 4.1.3

Multiply $- 1 (- \ln (x))$ .

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Step 4.1.3.1

Multiply $- 1$ by $- 1$ .

$- y \ln (x) + 1 \ln (x) = \ln (\frac{1}{x^{4}})$

Step 4.1.3.2

Multiply $\ln (x)$ by $1$ .

$- y \ln (x) + \ln (x) = \ln (\frac{1}{x^{4}})$

Step 5

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

$- y \ln (x) + \ln (x) - \ln (\frac{1}{x^{4}}) = 0$

Step 6

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$- y \ln (x) + \ln (\frac{x}{\frac{1}{x^{4}}}) = 0$

Step 7

Simplify each term.

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Step 7.1

Multiply the numerator by the reciprocal of the denominator.

$- y \ln (x) + \ln (x \cdot x^{4}) = 0$

Step 7.2

Multiply $x$ by $x^{4}$ by adding the exponents.

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Step 7.2.1

Multiply $x$ by $x^{4}$ .

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Step 7.2.1.1

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

$- y \ln (x) + \ln (x \cdot x^{4}) = 0$

Step 7.2.1.2

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

$- y \ln (x) + \ln (x^{1 + 4}) = 0$

Step 7.2.2

Add $1$ and $4$ .

$- y \ln (x) + \ln (x^{5}) = 0$

Step 8

Subtract $\ln (x^{5})$ from both sides of the equation.

$- y \ln (x) = - \ln (x^{5})$

Step 9

Divide each term in $- y \ln (x) = - \ln (x^{5})$ by $- \ln (x)$ and simplify.

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Step 9.1

Divide each term in $- y \ln (x) = - \ln (x^{5})$ by $- \ln (x)$ .

$\frac{- y \ln (x)}{- \ln (x)} = \frac{- \ln (x^{5})}{- \ln (x)}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Dividing two negative values results in a positive value.

$\frac{y \ln (x)}{\ln (x)} = \frac{- \ln (x^{5})}{- \ln (x)}$

Step 9.2.2

Cancel the common factor of $\ln (x)$ .

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Step 9.2.2.1

Cancel the common factor.

$\frac{y \ln (x)}{\ln (x)} = \frac{- \ln (x^{5})}{- \ln (x)}$

Step 9.2.2.2

Divide $y$ by $1$ .

$y = \frac{- \ln (x^{5})}{- \ln (x)}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Dividing two negative values results in a positive value.

$y = \frac{\ln (x^{5})}{\ln (x)}$