Finite Math Examples

$18000 = 3000 {(1 + \frac{0.06}{12})}^{12 t}$

Step 1

Rewrite the equation as $3000 {(1 + \frac{0.06}{12})}^{12 t} = 18000$ .

$3000 {(1 + \frac{0.06}{12})}^{12 t} = 18000$

Step 2

Simplify.

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Step 2.1

Divide $0.06$ by $12$ .

$3000 {(1 + 0.005)}^{12 t} = 18000$

Step 2.2

Add $1$ and $0.005$ .

$3000 \cdot {1.005}^{12 t} = 18000$

Step 3

Divide each term in $3000 \cdot {1.005}^{12 t} = 18000$ by $3000$ and simplify.

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Step 3.1

Divide each term in $3000 \cdot {1.005}^{12 t} = 18000$ by $3000$ .

$\frac{3000 \cdot {1.005}^{12 t}}{3000} = \frac{18000}{3000}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $3000$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{3000 \cdot {1.005}^{12 t}}{3000} = \frac{18000}{3000}$

Step 3.2.1.2

Divide ${1.005}^{12 t}$ by $1$ .

${1.005}^{12 t} = \frac{18000}{3000}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $18000$ by $3000$ .

${1.005}^{12 t} = 6$

Step 4

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

$\ln ({1.005}^{12 t}) = \ln (6)$

Step 5

Expand $\ln ({1.005}^{12 t})$ by moving $12 t$ outside the logarithm.

$12 t \ln (1.005) = \ln (6)$

Step 6

Divide each term in $12 t \ln (1.005) = \ln (6)$ by $12 \ln (1.005)$ and simplify.

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Step 6.1

Divide each term in $12 t \ln (1.005) = \ln (6)$ by $12 \ln (1.005)$ .

$\frac{12 t \ln (1.005)}{12 \ln (1.005)} = \frac{\ln (6)}{12 \ln (1.005)}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $12$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{12 t \ln (1.005)}{12 \ln (1.005)} = \frac{\ln (6)}{12 \ln (1.005)}$

Step 6.2.1.2

Rewrite the expression.

$\frac{t \ln (1.005)}{\ln (1.005)} = \frac{\ln (6)}{12 \ln (1.005)}$

Step 6.2.2

Cancel the common factor of $\ln (1.005)$ .

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Step 6.2.2.1

Cancel the common factor.

$\frac{t \ln (1.005)}{\ln (1.005)} = \frac{\ln (6)}{12 \ln (1.005)}$

Step 6.2.2.2

Divide $t$ by $1$ .

$t = \frac{\ln (6)}{12 \ln (1.005)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$t = \frac{\ln (6)}{12 \ln (1.005)}$

Decimal Form:

$t = 29.93725240 \dots$