Algebra Examples

$30000 = 640 {(1.085)}^{x + 1}$

Step 1

Rewrite the equation as $640 \cdot {1.085}^{x + 1} = 30000$ .

$640 \cdot {1.085}^{x + 1} = 30000$

Step 2

Divide each term in $640 \cdot {1.085}^{x + 1} = 30000$ by $640$ and simplify.

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Step 2.1

Divide each term in $640 \cdot {1.085}^{x + 1} = 30000$ by $640$ .

$\frac{640 \cdot {1.085}^{x + 1}}{640} = \frac{30000}{640}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $640$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{640 \cdot {1.085}^{x + 1}}{640} = \frac{30000}{640}$

Step 2.2.1.2

Divide ${1.085}^{x + 1}$ by $1$ .

${1.085}^{x + 1} = \frac{30000}{640}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $30000$ and $640$ .

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Step 2.3.1.1

Factor $80$ out of $30000$ .

${1.085}^{x + 1} = \frac{80 (375)}{640}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $80$ out of $640$ .

${1.085}^{x + 1} = \frac{80 \cdot 375}{80 \cdot 8}$

Step 2.3.1.2.2

Cancel the common factor.

${1.085}^{x + 1} = \frac{80 \cdot 375}{80 \cdot 8}$

Step 2.3.1.2.3

Rewrite the expression.

${1.085}^{x + 1} = \frac{375}{8}$

Step 3

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

$\ln ({1.085}^{x + 1}) = \ln (\frac{375}{8})$

Step 4

Expand $\ln ({1.085}^{x + 1})$ by moving $x + 1$ outside the logarithm.

$(x + 1) \ln (1.085) = \ln (\frac{375}{8})$

Step 5

Simplify the left side.

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Step 5.1

Simplify $(x + 1) \ln (1.085)$ .

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Step 5.1.1

Apply the distributive property.

$x \ln (1.085) + 1 \ln (1.085) = \ln (\frac{375}{8})$

Step 5.1.2

Multiply $\ln (1.085)$ by $1$ .

$x \ln (1.085) + \ln (1.085) = \ln (\frac{375}{8})$

Step 6

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

$x \ln (1.085) + \ln (1.085) - \ln (\frac{375}{8}) = 0$

Step 7

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

$x \ln (1.085) + \ln (\frac{1.085}{\frac{375}{8}}) = 0$

Step 8

Simplify each term.

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Step 8.1

Multiply the numerator by the reciprocal of the denominator.

$x \ln (1.085) + \ln (1.085 (\frac{8}{375})) = 0$

Step 8.2

Multiply $1.085 (\frac{8}{375})$ .

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Step 8.2.1

Combine $1.085$ and $\frac{8}{375}$ .

$x \ln (1.085) + \ln (\frac{1.085 \cdot 8}{375}) = 0$

Step 8.2.2

Multiply $1.085$ by $8$ .

$x \ln (1.085) + \ln (\frac{8.68}{375}) = 0$

Step 8.3

Divide $8.68$ by $375$ .

$x \ln (1.085) + \ln (0.02314\overline{6}) = 0$

Step 9

Subtract $\ln (0.02314\overline{6})$ from both sides of the equation.

$x \ln (1.085) = - \ln (0.02314\overline{6})$

Step 10

Divide each term in $x \ln (1.085) = - \ln (0.02314\overline{6})$ by $\ln (1.085)$ and simplify.

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Step 10.1

Divide each term in $x \ln (1.085) = - \ln (0.02314\overline{6})$ by $\ln (1.085)$ .

$\frac{x \ln (1.085)}{\ln (1.085)} = \frac{- \ln (0.02314\overline{6})}{\ln (1.085)}$

Step 10.2

Simplify the left side.

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Step 10.2.1

Cancel the common factor of $\ln (1.085)$ .

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Step 10.2.1.1

Cancel the common factor.

$\frac{x \ln (1.085)}{\ln (1.085)} = \frac{- \ln (0.02314\overline{6})}{\ln (1.085)}$

Step 10.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \ln (0.02314\overline{6})}{\ln (1.085)}$

Step 10.3

Simplify the right side.

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Step 10.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (0.02314\overline{6})}{\ln (1.085)}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (0.02314\overline{6})}{\ln (1.085)}$

Decimal Form:

$x = 46.16211200 \dots$