Pre-Algebra Examples

${0.25}^{x + 2} = a^{- 2}$

Step 1

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

$\ln ({0.25}^{x + 2}) = \ln (a^{- 2})$

Step 2

Expand $\ln ({0.25}^{x + 2})$ by moving $x + 2$ outside the logarithm.

$(x + 2) \ln (0.25) = \ln (a^{- 2})$

Step 3

Simplify the left side.

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Step 3.1

Apply the distributive property.

$x \ln (0.25) + 2 \ln (0.25) = \ln (a^{- 2})$

Step 4

Simplify the right side.

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Step 4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x \ln (0.25) + 2 \ln (0.25) = \ln (\frac{1}{a^{2}})$

Step 5

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

$x \ln (0.25) + 2 \ln (0.25) - \ln (\frac{1}{a^{2}}) = 0$

Step 6

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

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Step 6.1

Subtract $2 \ln (0.25)$ from both sides of the equation.

$x \ln (0.25) - \ln (\frac{1}{a^{2}}) = - 2 \ln (0.25)$

Step 6.2

Add $\ln (\frac{1}{a^{2}})$ to both sides of the equation.

$x \ln (0.25) = - 2 \ln (0.25) + \ln (\frac{1}{a^{2}})$

Step 7

Divide each term in $x \ln (0.25) = - 2 \ln (0.25) + \ln (\frac{1}{a^{2}})$ by $\ln (0.25)$ and simplify.

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Step 7.1

Divide each term in $x \ln (0.25) = - 2 \ln (0.25) + \ln (\frac{1}{a^{2}})$ by $\ln (0.25)$ .

$\frac{x \ln (0.25)}{\ln (0.25)} = \frac{- 2 \ln (0.25)}{\ln (0.25)} + \frac{\ln (\frac{1}{a^{2}})}{\ln (0.25)}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $\ln (0.25)$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{x \ln (0.25)}{\ln (0.25)} = \frac{- 2 \ln (0.25)}{\ln (0.25)} + \frac{\ln (\frac{1}{a^{2}})}{\ln (0.25)}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2 \ln (0.25)}{\ln (0.25)} + \frac{\ln (\frac{1}{a^{2}})}{\ln (0.25)}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Cancel the common factor of $\ln (0.25)$ .

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Step 7.3.1.1

Cancel the common factor.

$x = \frac{- 2 \ln (0.25)}{\ln (0.25)} + \frac{\ln (\frac{1}{a^{2}})}{\ln (0.25)}$

Step 7.3.1.2

Divide $- 2$ by $1$ .

$x = - 2 + \frac{\ln (\frac{1}{a^{2}})}{\ln (0.25)}$