Precalculus Examples

${(2.221)}^{x} = (\frac{1}{2}) ({0.75}^{x + 2})$

Step 1

Take the log of both sides of the equation.

$\ln ({2.221}^{x}) = \ln (\frac{1}{2} \cdot {0.75}^{x + 2})$

Step 2

Expand $\ln ({2.221}^{x})$ by moving $x$ outside the logarithm.

$x \ln (2.221) = \ln (\frac{1}{2} \cdot {0.75}^{x + 2})$

Step 3

Rewrite $\ln (\frac{1}{2} \cdot {0.75}^{x + 2})$ as $\ln (\frac{1}{2}) + \ln ({0.75}^{x + 2})$ .

$x \ln (2.221) = \ln (\frac{1}{2}) + \ln ({0.75}^{x + 2})$

Step 4

Rewrite $\ln (\frac{1}{2})$ as $\ln (1) - \ln (2)$ .

$x \ln (2.221) = \ln (1) - \ln (2) + \ln ({0.75}^{x + 2})$

Step 5

Expand $\ln ({0.75}^{x + 2})$ by moving $x + 2$ outside the logarithm.

$x \ln (2.221) = \ln (1) - \ln (2) + (x + 2) \ln (0.75)$

Step 6

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

$x \ln (2.221) = 0 - \ln (2) + (x + 2) \ln (0.75)$

Step 7

Subtract $\ln (2)$ from $0$ .

$x \ln (2.221) = - \ln (2) + (x + 2) \ln (0.75)$

Step 8

Solve the equation for $x$ .

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Step 8.1

Simplify the right side.

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Step 8.1.1

Simplify each term.

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Step 8.1.1.1

Apply the distributive property.

$x \ln (2.221) = - \ln (2) + x \ln (0.75) + 2 \ln (0.75)$

Step 8.1.1.2

Simplify $2 \ln (0.75)$ by moving $2$ inside the logarithm.

$x \ln (2.221) = - \ln (2) + x \ln (0.75) + \ln ({0.75}^{2})$

Step 8.1.1.3

Raise $0.75$ to the power of $2$ .

$x \ln (2.221) = - \ln (2) + x \ln (0.75) + \ln (0.5625)$

Step 8.2

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

$x \ln (2.221) + \ln (2) - x \ln (0.75) - \ln (0.5625) = 0$

Step 8.3

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

$x \ln (2.221) - x \ln (0.75) + \ln (\frac{2}{0.5625}) = 0$

Step 8.4

Divide $2$ by $0.5625$ .

$x \ln (2.221) - x \ln (0.75) + \ln (3.\overline{5}) = 0$

Step 8.5

Subtract $\ln (3.\overline{5})$ from both sides of the equation.

$x \ln (2.221) - x \ln (0.75) = - \ln (3.\overline{5})$

Step 8.6

Factor $x$ out of $x \ln (2.221) - x \ln (0.75)$ .

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Step 8.6.1

Factor $x$ out of $x \ln (2.221)$ .

$x (\ln (2.221)) - x \ln (0.75) = - \ln (3.\overline{5})$

Step 8.6.2

Factor $x$ out of $- x \ln (0.75)$ .

$x (\ln (2.221)) + x (- 1 \ln (0.75)) = - \ln (3.\overline{5})$

Step 8.6.3

Factor $x$ out of $x (\ln (2.221)) + x (- 1 \ln (0.75))$ .

$x (\ln (2.221) - 1 \ln (0.75)) = - \ln (3.\overline{5})$

Step 8.7

Rewrite $- 1 \ln (0.75)$ as $- \ln (0.75)$ .

$x (\ln (2.221) - \ln (0.75)) = - \ln (3.\overline{5})$

Step 8.8

Divide each term in $x (\ln (2.221) - \ln (0.75)) = - \ln (3.\overline{5})$ by $\ln (2.221) - \ln (0.75)$ and simplify.

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Step 8.8.1

Divide each term in $x (\ln (2.221) - \ln (0.75)) = - \ln (3.\overline{5})$ by $\ln (2.221) - \ln (0.75)$ .

$\frac{x (\ln (2.221) - \ln (0.75))}{\ln (2.221) - \ln (0.75)} = \frac{- \ln (3.\overline{5})}{\ln (2.221) - \ln (0.75)}$

Step 8.8.2

Simplify the left side.

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Step 8.8.2.1

Cancel the common factor of $\ln (2.221) - \ln (0.75)$ .

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Step 8.8.2.1.1

Cancel the common factor.

$\frac{x (\ln (2.221) - \ln (0.75))}{\ln (2.221) - \ln (0.75)} = \frac{- \ln (3.\overline{5})}{\ln (2.221) - \ln (0.75)}$

Step 8.8.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \ln (3.\overline{5})}{\ln (2.221) - \ln (0.75)}$

Step 8.8.3

Simplify the right side.

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Step 8.8.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (3.\overline{5})}{\ln (2.221) - \ln (0.75)}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (3.\overline{5})}{\ln (2.221) - \ln (0.75)}$

Decimal Form:

$x = - 1.16844605 \dots$