Finite Math Examples

$4000 = 2000 {(1 + \frac{0.09}{4})}^{4 x}$

Step 1

Rewrite the equation as $2000 {(1 + \frac{0.09}{4})}^{4 x} = 4000$ .

$2000 {(1 + \frac{0.09}{4})}^{4 x} = 4000$

Step 2

Simplify.

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Step 2.1

Divide $0.09$ by $4$ .

$2000 {(1 + 0.0225)}^{4 x} = 4000$

Step 2.2

Add $1$ and $0.0225$ .

$2000 \cdot {1.0225}^{4 x} = 4000$

Step 3

Divide each term in $2000 \cdot {1.0225}^{4 x} = 4000$ by $2000$ and simplify.

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Step 3.1

Divide each term in $2000 \cdot {1.0225}^{4 x} = 4000$ by $2000$ .

$\frac{2000 \cdot {1.0225}^{4 x}}{2000} = \frac{4000}{2000}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2000$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2000 \cdot {1.0225}^{4 x}}{2000} = \frac{4000}{2000}$

Step 3.2.1.2

Divide ${1.0225}^{4 x}$ by $1$ .

${1.0225}^{4 x} = \frac{4000}{2000}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $4000$ by $2000$ .

${1.0225}^{4 x} = 2$

Step 4

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

$\ln ({1.0225}^{4 x}) = \ln (2)$

Step 5

Expand $\ln ({1.0225}^{4 x})$ by moving $4 x$ outside the logarithm.

$4 x \ln (1.0225) = \ln (2)$

Step 6

Divide each term in $4 x \ln (1.0225) = \ln (2)$ by $4 \ln (1.0225)$ and simplify.

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Step 6.1

Divide each term in $4 x \ln (1.0225) = \ln (2)$ by $4 \ln (1.0225)$ .

$\frac{4 x \ln (1.0225)}{4 \ln (1.0225)} = \frac{\ln (2)}{4 \ln (1.0225)}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $4$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{4 x \ln (1.0225)}{4 \ln (1.0225)} = \frac{\ln (2)}{4 \ln (1.0225)}$

Step 6.2.1.2

Rewrite the expression.

$\frac{x \ln (1.0225)}{\ln (1.0225)} = \frac{\ln (2)}{4 \ln (1.0225)}$

Step 6.2.2

Cancel the common factor of $\ln (1.0225)$ .

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Step 6.2.2.1

Cancel the common factor.

$\frac{x \ln (1.0225)}{\ln (1.0225)} = \frac{\ln (2)}{4 \ln (1.0225)}$

Step 6.2.2.2

Divide $x$ by $1$ .

$x = \frac{\ln (2)}{4 \ln (1.0225)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (2)}{4 \ln (1.0225)}$

Decimal Form:

$x = 7.78795742 \dots$