Pre-Algebra Examples

$(2^{x}) (3^{y}) (5^{z}) = (2^{3}) (3^{x - 2}) (5^{2 x - 3 y})$

Step 1

Divide each term in $2^{x} \cdot 3^{y} \cdot 5^{z} = 2^{3} \cdot 3^{x - 2} \cdot 5^{2 x - 3 y}$ by $2^{x} \cdot 3^{y}$ and simplify.

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Step 1.1

Divide each term in $2^{x} \cdot 3^{y} \cdot 5^{z} = 2^{3} \cdot 3^{x - 2} \cdot 5^{2 x - 3 y}$ by $2^{x} \cdot 3^{y}$ .

$\frac{2^{x} \cdot 3^{y} \cdot 5^{z}}{2^{x} \cdot 3^{y}} = \frac{2^{3} \cdot 3^{x - 2} \cdot 5^{2 x - 3 y}}{2^{x} \cdot 3^{y}}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $2^{x}$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{2^{x} \cdot 3^{y} \cdot 5^{z}}{2^{x} \cdot 3^{y}} = \frac{2^{3} \cdot 3^{x - 2} \cdot 5^{2 x - 3 y}}{2^{x} \cdot 3^{y}}$

Step 1.2.1.2

Rewrite the expression.

$\frac{3^{y} \cdot 5^{z}}{3^{y}} = \frac{2^{3} \cdot 3^{x - 2} \cdot 5^{2 x - 3 y}}{2^{x} \cdot 3^{y}}$

Step 1.2.2

Cancel the common factor of $3^{y}$ .

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Step 1.2.2.1

Cancel the common factor.

$\frac{3^{y} \cdot 5^{z}}{3^{y}} = \frac{2^{3} \cdot 3^{x - 2} \cdot 5^{2 x - 3 y}}{2^{x} \cdot 3^{y}}$

Step 1.2.2.2

Divide $5^{z}$ by $1$ .

$5^{z} = \frac{2^{3} \cdot 3^{x - 2} \cdot 5^{2 x - 3 y}}{2^{x} \cdot 3^{y}}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Cancel the common factor of $2^{3}$ and $2^{x}$ .

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Step 1.3.1.1

Factor $2^{x}$ out of $2^{3} \cdot 3^{x - 2} \cdot 5^{2 x - 3 y}$ .

$5^{z} = \frac{2^{x} (2^{3 - x} \cdot 3^{x - 2} \cdot 5^{2 x - 3 y})}{2^{x} \cdot 3^{y}}$

Step 1.3.1.2

Cancel the common factors.

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Step 1.3.1.2.1

Factor $2^{x}$ out of $2^{x} \cdot 3^{y}$ .

$5^{z} = \frac{2^{x} (2^{3 - x} \cdot 3^{x - 2} \cdot 5^{2 x - 3 y})}{2^{x} (3^{y})}$

Step 1.3.1.2.2

Cancel the common factor.

$5^{z} = \frac{2^{x} (2^{3 - x} \cdot 3^{x - 2} \cdot 5^{2 x - 3 y})}{2^{x} \cdot 3^{y}}$

Step 1.3.1.2.3

Rewrite the expression.

$5^{z} = \frac{2^{3 - x} \cdot 3^{x - 2} \cdot 5^{2 x - 3 y}}{3^{y}}$

Step 1.3.2

Cancel the common factor of $3^{x - 2}$ and $3^{y}$ .

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Step 1.3.2.1

Factor $3^{y}$ out of $2^{3 - x} \cdot 3^{x - 2} \cdot 5^{2 x - 3 y}$ .

$5^{z} = \frac{3^{y} (2^{3 - x} \cdot 3^{x - 2 - y} \cdot 5^{2 x - 3 y})}{3^{y}}$

Step 1.3.2.2

Cancel the common factors.

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Step 1.3.2.2.1

Multiply by $1$ .

$5^{z} = \frac{3^{y} (2^{3 - x} \cdot 3^{x - 2 - y} \cdot 5^{2 x - 3 y})}{3^{y} \cdot 1}$

Step 1.3.2.2.2

Cancel the common factor.

$5^{z} = \frac{3^{y} (2^{3 - x} \cdot 3^{x - 2 - y} \cdot 5^{2 x - 3 y})}{3^{y} \cdot 1}$

Step 1.3.2.2.3

Rewrite the expression.

$5^{z} = \frac{2^{3 - x} \cdot 3^{x - 2 - y} \cdot 5^{2 x - 3 y}}{1}$

Step 1.3.2.2.4

Divide $2^{3 - x} \cdot 3^{x - 2 - y} \cdot 5^{2 x - 3 y}$ by $1$ .

$5^{z} = 2^{3 - x} \cdot 3^{x - 2 - y} \cdot 5^{2 x - 3 y}$

Step 2

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

$\ln (5^{z}) = \ln (2^{3 - x} \cdot 3^{x - 2 - y} \cdot 5^{2 x - 3 y})$

Step 3

Expand $\ln (5^{z})$ by moving $z$ outside the logarithm.

$z \ln (5) = \ln (2^{3 - x} \cdot 3^{x - 2 - y} \cdot 5^{2 x - 3 y})$

Step 4

Expand the right side.

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Step 4.1

Rewrite $\ln (2^{3 - x} \cdot 3^{x - 2 - y} \cdot 5^{2 x - 3 y})$ as $\ln (2^{3 - x} \cdot 3^{x - 2 - y}) + \ln (5^{2 x - 3 y})$ .

$z \ln (5) = \ln (2^{3 - x} \cdot 3^{x - 2 - y}) + \ln (5^{2 x - 3 y})$

Step 4.2

Rewrite $\ln (2^{3 - x} \cdot 3^{x - 2 - y})$ as $\ln (2^{3 - x}) + \ln (3^{x - 2 - y})$ .

$z \ln (5) = \ln (2^{3 - x}) + \ln (3^{x - 2 - y}) + \ln (5^{2 x - 3 y})$

Step 4.3

Expand $\ln (2^{3 - x})$ by moving $3 - x$ outside the logarithm.

$z \ln (5) = (3 - x) \ln (2) + \ln (3^{x - 2 - y}) + \ln (5^{2 x - 3 y})$

Step 4.4

Expand $\ln (3^{x - 2 - y})$ by moving $x - 2 - y$ outside the logarithm.

$z \ln (5) = (3 - x) \ln (2) + (x - 2 - y) \ln (3) + \ln (5^{2 x - 3 y})$

Step 4.5

Expand $\ln (5^{2 x - 3 y})$ by moving $2 x - 3 y$ outside the logarithm.

$z \ln (5) = (3 - x) \ln (2) + (x - 2 - y) \ln (3) + (2 x - 3 y) \ln (5)$

Step 5

Simplify the right side.

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Step 5.1

Simplify each term.

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Step 5.1.1

Apply the distributive property.

$z \ln (5) = 3 \ln (2) - x \ln (2) + (x - 2 - y) \ln (3) + (2 x - 3 y) \ln (5)$

Step 5.1.2

Apply the distributive property.

$z \ln (5) = 3 \ln (2) - x \ln (2) + x \ln (3) - 2 \ln (3) - y \ln (3) + (2 x - 3 y) \ln (5)$

Step 5.1.3

Apply the distributive property.

$z \ln (5) = 3 \ln (2) - x \ln (2) + x \ln (3) - 2 \ln (3) - y \ln (3) + 2 x \ln (5) - 3 y \ln (5)$

Step 6

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

$z \ln (5) - 3 \ln (2) + x \ln (2) - x \ln (3) + 2 \ln (3) + y \ln (3) - 2 x \ln (5) + 3 y \ln (5) = 0$

Step 7

Move all terms not containing $z$ to the right side of the equation.

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Step 7.1

Add $3 \ln (2)$ to both sides of the equation.

$z \ln (5) + x \ln (2) - x \ln (3) + 2 \ln (3) + y \ln (3) - 2 x \ln (5) + 3 y \ln (5) = 3 \ln (2)$

Step 7.2

Subtract $x \ln (2)$ from both sides of the equation.

$z \ln (5) - x \ln (3) + 2 \ln (3) + y \ln (3) - 2 x \ln (5) + 3 y \ln (5) = 3 \ln (2) - x \ln (2)$

Step 7.3

Add $x \ln (3)$ to both sides of the equation.

$z \ln (5) + 2 \ln (3) + y \ln (3) - 2 x \ln (5) + 3 y \ln (5) = 3 \ln (2) - x \ln (2) + x \ln (3)$

Step 7.4

Subtract $2 \ln (3)$ from both sides of the equation.

$z \ln (5) + y \ln (3) - 2 x \ln (5) + 3 y \ln (5) = 3 \ln (2) - x \ln (2) + x \ln (3) - 2 \ln (3)$

Step 7.5

Subtract $y \ln (3)$ from both sides of the equation.

$z \ln (5) - 2 x \ln (5) + 3 y \ln (5) = 3 \ln (2) - x \ln (2) + x \ln (3) - 2 \ln (3) - y \ln (3)$

Step 7.6

Add $2 x \ln (5)$ to both sides of the equation.

$z \ln (5) + 3 y \ln (5) = 3 \ln (2) - x \ln (2) + x \ln (3) - 2 \ln (3) - y \ln (3) + 2 x \ln (5)$

Step 7.7

Subtract $3 y \ln (5)$ from both sides of the equation.

$z \ln (5) = 3 \ln (2) - x \ln (2) + x \ln (3) - 2 \ln (3) - y \ln (3) + 2 x \ln (5) - 3 y \ln (5)$

Step 8

Divide each term in $z \ln (5) = 3 \ln (2) - x \ln (2) + x \ln (3) - 2 \ln (3) - y \ln (3) + 2 x \ln (5) - 3 y \ln (5)$ by $\ln (5)$ and simplify.

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Step 8.1

Divide each term in $z \ln (5) = 3 \ln (2) - x \ln (2) + x \ln (3) - 2 \ln (3) - y \ln (3) + 2 x \ln (5) - 3 y \ln (5)$ by $\ln (5)$ .

$\frac{z \ln (5)}{\ln (5)} = \frac{3 \ln (2)}{\ln (5)} + \frac{- x \ln (2)}{\ln (5)} + \frac{x \ln (3)}{\ln (5)} + \frac{- 2 \ln (3)}{\ln (5)} + \frac{- y \ln (3)}{\ln (5)} + \frac{2 x \ln (5)}{\ln (5)} + \frac{- 3 y \ln (5)}{\ln (5)}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $\ln (5)$ .

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Step 8.2.1.1

Cancel the common factor.

Step 8.2.1.2

Divide $z$ by $1$ .

$z = \frac{3 \ln (2)}{\ln (5)} + \frac{- x \ln (2)}{\ln (5)} + \frac{x \ln (3)}{\ln (5)} + \frac{- 2 \ln (3)}{\ln (5)} + \frac{- y \ln (3)}{\ln (5)} + \frac{2 x \ln (5)}{\ln (5)} + \frac{- 3 y \ln (5)}{\ln (5)}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Simplify each term.

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Step 8.3.1.1

Move the negative in front of the fraction.

$z = \frac{3 \ln (2)}{\ln (5)} - \frac{x \ln (2)}{\ln (5)} + \frac{x \ln (3)}{\ln (5)} + \frac{- 2 \ln (3)}{\ln (5)} + \frac{- y \ln (3)}{\ln (5)} + \frac{2 x \ln (5)}{\ln (5)} + \frac{- 3 y \ln (5)}{\ln (5)}$

Step 8.3.1.2

Move the negative in front of the fraction.

$z = \frac{3 \ln (2)}{\ln (5)} - \frac{x \ln (2)}{\ln (5)} + \frac{x \ln (3)}{\ln (5)} - \frac{2 \ln (3)}{\ln (5)} + \frac{- y \ln (3)}{\ln (5)} + \frac{2 x \ln (5)}{\ln (5)} + \frac{- 3 y \ln (5)}{\ln (5)}$

Step 8.3.1.3

Move the negative in front of the fraction.

$z = \frac{3 \ln (2)}{\ln (5)} - \frac{x \ln (2)}{\ln (5)} + \frac{x \ln (3)}{\ln (5)} - \frac{2 \ln (3)}{\ln (5)} - \frac{y \ln (3)}{\ln (5)} + \frac{2 x \ln (5)}{\ln (5)} + \frac{- 3 y \ln (5)}{\ln (5)}$

Step 8.3.1.4

Cancel the common factor of $\ln (5)$ .

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Step 8.3.1.4.1

Cancel the common factor.

Step 8.3.1.4.2

Divide $2 x$ by $1$ .

$z = \frac{3 \ln (2)}{\ln (5)} - \frac{x \ln (2)}{\ln (5)} + \frac{x \ln (3)}{\ln (5)} - \frac{2 \ln (3)}{\ln (5)} - \frac{y \ln (3)}{\ln (5)} + 2 x + \frac{- 3 y \ln (5)}{\ln (5)}$

Step 8.3.1.5

Cancel the common factor of $\ln (5)$ .

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Step 8.3.1.5.1

Cancel the common factor.

$z = \frac{3 \ln (2)}{\ln (5)} - \frac{x \ln (2)}{\ln (5)} + \frac{x \ln (3)}{\ln (5)} - \frac{2 \ln (3)}{\ln (5)} - \frac{y \ln (3)}{\ln (5)} + 2 x + \frac{- 3 y \ln (5)}{\ln (5)}$

Step 8.3.1.5.2

Divide $- 3 y$ by $1$ .

$z = \frac{3 \ln (2)}{\ln (5)} - \frac{x \ln (2)}{\ln (5)} + \frac{x \ln (3)}{\ln (5)} - \frac{2 \ln (3)}{\ln (5)} - \frac{y \ln (3)}{\ln (5)} + 2 x - 3 y$