Pre-Algebra Examples

$\frac{5}{3^{x}} = \frac{27}{125}$

Step 1

Multiply both sides by $3^{x}$ .

$\frac{5}{3^{x}} \cdot 3^{x} = \frac{27}{125} \cdot 3^{x}$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

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

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Step 2.1.1.1

Cancel the common factor.

$\frac{5}{3^{x}} \cdot 3^{x} = \frac{27}{125} \cdot 3^{x}$

Step 2.1.1.2

Rewrite the expression.

$5 = \frac{27}{125} \cdot 3^{x}$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $\frac{27}{125} \cdot 3^{x}$ .

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Step 2.2.1.1

Combine $\frac{27}{125}$ and $3^{x}$ .

$5 = \frac{27 \cdot 3^{x}}{125}$

Step 2.2.1.2

Rewrite $27$ as $3^{3}$ .

$5 = \frac{3^{3} \cdot 3^{x}}{125}$

Step 2.2.1.3

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

$5 = \frac{3^{3 + x}}{125}$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $\frac{3^{3 + x}}{125} = 5$ .

$\frac{3^{3 + x}}{125} = 5$

Step 3.2

Multiply both sides of the equation by $125$ .

$125 \frac{3^{3 + x}}{125} = 125 \cdot 5$

Step 3.3

Simplify both sides of the equation.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Cancel the common factor of $125$ .

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Step 3.3.1.1.1

Cancel the common factor.

$125 \frac{3^{3 + x}}{125} = 125 \cdot 5$

Step 3.3.1.1.2

Rewrite the expression.

$3^{3 + x} = 125 \cdot 5$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Multiply $125$ by $5$ .

$3^{3 + x} = 625$

Step 3.4

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

$\ln (3^{3 + x}) = \ln (625)$

Step 3.5

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

$(3 + x) \ln (3) = \ln (625)$

Step 3.6

Simplify the left side.

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Step 3.6.1

Apply the distributive property.

$3 \ln (3) + x \ln (3) = \ln (625)$

Step 3.7

Reorder $3 \ln (3)$ and $x \ln (3)$ .

$x \ln (3) + 3 \ln (3) = \ln (625)$

Step 3.8

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

$x \ln (3) + 3 \ln (3) - \ln (625) = 0$

Step 3.9

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

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Step 3.9.1

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

$x \ln (3) - \ln (625) = - 3 \ln (3)$

Step 3.9.2

Add $\ln (625)$ to both sides of the equation.

$x \ln (3) = - 3 \ln (3) + \ln (625)$

Step 3.10

Divide each term in $x \ln (3) = - 3 \ln (3) + \ln (625)$ by $\ln (3)$ and simplify.

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Step 3.10.1

Divide each term in $x \ln (3) = - 3 \ln (3) + \ln (625)$ by $\ln (3)$ .

$\frac{x \ln (3)}{\ln (3)} = \frac{- 3 \ln (3)}{\ln (3)} + \frac{\ln (625)}{\ln (3)}$

Step 3.10.2

Simplify the left side.

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Step 3.10.2.1

Cancel the common factor of $\ln (3)$ .

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Step 3.10.2.1.1

Cancel the common factor.

$\frac{x \ln (3)}{\ln (3)} = \frac{- 3 \ln (3)}{\ln (3)} + \frac{\ln (625)}{\ln (3)}$

Step 3.10.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3 \ln (3)}{\ln (3)} + \frac{\ln (625)}{\ln (3)}$

Step 3.10.3

Simplify the right side.

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Step 3.10.3.1

Cancel the common factor of $\ln (3)$ .

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Step 3.10.3.1.1

Cancel the common factor.

$x = \frac{- 3 \ln (3)}{\ln (3)} + \frac{\ln (625)}{\ln (3)}$

Step 3.10.3.1.2

Divide $- 3$ by $1$ .

$x = - 3 + \frac{\ln (625)}{\ln (3)}$