Algebra Examples

$1206 = 8600 {(0.96)}^{x + 1} - 1194$

Step 1

Rewrite the equation as $8600 \cdot {0.96}^{x + 1} - 1194 = 1206$ .

$8600 \cdot {0.96}^{x + 1} - 1194 = 1206$

Step 2

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

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Step 2.1

Add $1194$ to both sides of the equation.

$8600 \cdot {0.96}^{x + 1} = 1206 + 1194$

Step 2.2

Add $1206$ and $1194$ .

$8600 \cdot {0.96}^{x + 1} = 2400$

Step 3

Divide each term in $8600 \cdot {0.96}^{x + 1} = 2400$ by $8600$ and simplify.

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Step 3.1

Divide each term in $8600 \cdot {0.96}^{x + 1} = 2400$ by $8600$ .

$\frac{8600 \cdot {0.96}^{x + 1}}{8600} = \frac{2400}{8600}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $8600$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{8600 \cdot {0.96}^{x + 1}}{8600} = \frac{2400}{8600}$

Step 3.2.1.2

Divide ${0.96}^{x + 1}$ by $1$ .

${0.96}^{x + 1} = \frac{2400}{8600}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $2400$ and $8600$ .

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Step 3.3.1.1

Factor $200$ out of $2400$ .

${0.96}^{x + 1} = \frac{200 (12)}{8600}$

Step 3.3.1.2

Cancel the common factors.

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Step 3.3.1.2.1

Factor $200$ out of $8600$ .

${0.96}^{x + 1} = \frac{200 \cdot 12}{200 \cdot 43}$

Step 3.3.1.2.2

Cancel the common factor.

${0.96}^{x + 1} = \frac{200 \cdot 12}{200 \cdot 43}$

Step 3.3.1.2.3

Rewrite the expression.

${0.96}^{x + 1} = \frac{12}{43}$

Step 4

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

$\ln ({0.96}^{x + 1}) = \ln (\frac{12}{43})$

Step 5

Expand $\ln ({0.96}^{x + 1})$ by moving $x + 1$ outside the logarithm.

$(x + 1) \ln (0.96) = \ln (\frac{12}{43})$

Step 6

Simplify the left side.

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Step 6.1

Simplify $(x + 1) \ln (0.96)$ .

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Step 6.1.1

Apply the distributive property.

$x \ln (0.96) + 1 \ln (0.96) = \ln (\frac{12}{43})$

Step 6.1.2

Multiply $\ln (0.96)$ by $1$ .

$x \ln (0.96) + \ln (0.96) = \ln (\frac{12}{43})$

Step 7

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

$x \ln (0.96) + \ln (0.96) - \ln (\frac{12}{43}) = 0$

Step 8

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

$x \ln (0.96) + \ln (\frac{0.96}{\frac{12}{43}}) = 0$

Step 9

Simplify each term.

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Step 9.1

Multiply the numerator by the reciprocal of the denominator.

$x \ln (0.96) + \ln (0.96 (\frac{43}{12})) = 0$

Step 9.2

Cancel the common factor of $0.48$ .

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Step 9.2.1

Factor $0.48$ out of $0.96$ .

$x \ln (0.96) + \ln (0.48 (2) (\frac{43}{12})) = 0$

Step 9.2.2

Factor $0.48$ out of $12$ .

$x \ln (0.96) + \ln (0.48 \cdot (2 (\frac{43}{0.48 \cdot 25}))) = 0$

Step 9.2.3

Cancel the common factor.

$x \ln (0.96) + \ln (0.48 \cdot (2 (\frac{43}{0.48 \cdot 25}))) = 0$

Step 9.2.4

Rewrite the expression.

$x \ln (0.96) + \ln (2 (\frac{43}{25})) = 0$

Step 9.3

Combine $2$ and $\frac{43}{25}$ .

$x \ln (0.96) + \ln (\frac{2 \cdot 43}{25}) = 0$

Step 9.4

Multiply $2$ by $43$ .

$x \ln (0.96) + \ln (\frac{86}{25}) = 0$

Step 10

Subtract $\ln (\frac{86}{25})$ from both sides of the equation.

$x \ln (0.96) = - \ln (\frac{86}{25})$

Step 11

Divide each term in $x \ln (0.96) = - \ln (\frac{86}{25})$ by $\ln (0.96)$ and simplify.

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Step 11.1

Divide each term in $x \ln (0.96) = - \ln (\frac{86}{25})$ by $\ln (0.96)$ .

$\frac{x \ln (0.96)}{\ln (0.96)} = \frac{- \ln (\frac{86}{25})}{\ln (0.96)}$

Step 11.2

Simplify the left side.

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Step 11.2.1

Cancel the common factor of $\ln (0.96)$ .

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Step 11.2.1.1

Cancel the common factor.

$\frac{x \ln (0.96)}{\ln (0.96)} = \frac{- \ln (\frac{86}{25})}{\ln (0.96)}$

Step 11.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \ln (\frac{86}{25})}{\ln (0.96)}$

Step 11.3

Simplify the right side.

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Step 11.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (\frac{86}{25})}{\ln (0.96)}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (\frac{86}{25})}{\ln (0.96)}$

Decimal Form:

$x = 30.26484829 \dots$