Basic Math Examples

2000 {(1 + \frac{0.06}{12})}^{12 t} = 10000

$2000 {(1 + \frac{0.06}{12})}^{12 t} = 10000$

Step 1

Simplify.

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Step 1.1

Divide

0.06

$0.06$ by

12

$12$ .

2000 {(1 + 0.005)}^{12 t} = 10000

$2000 {(1 + 0.005)}^{12 t} = 10000$

Step 1.2

Add

1

$1$ and

0.005

$0.005$ .

2000 \cdot {1.005}^{12 t} = 10000

$2000 \cdot {1.005}^{12 t} = 10000$

2000 \cdot {1.005}^{12 t} = 10000

$2000 \cdot {1.005}^{12 t} = 10000$

Step 2

Divide each term in

2000 \cdot {1.005}^{12 t} = 10000

$2000 \cdot {1.005}^{12 t} = 10000$ by

2000

$2000$ and simplify.

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Step 2.1

Divide each term in

2000 \cdot {1.005}^{12 t} = 10000

$2000 \cdot {1.005}^{12 t} = 10000$ by

2000

$2000$ .

\frac{2000 \cdot {1.005}^{12 t}}{2000} = \frac{10000}{2000}

$\frac{2000 \cdot {1.005}^{12 t}}{2000} = \frac{10000}{2000}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of

2000

$2000$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2000 \cdot {1.005}^{12 t}}{2000} = \frac{10000}{2000}$

Step 2.2.1.2

Divide

${1.005}^{12 t}$ by

$1$ .

${1.005}^{12 t} = \frac{10000}{2000}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide

$10000$ by

$2000$ .

${1.005}^{12 t} = 5$

Step 3

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

$\ln ({1.005}^{12 t}) = \ln (5)$

Step 4

Expand

$\ln ({1.005}^{12 t})$ by moving

$12 t$ outside the logarithm.

$12 t \ln (1.005) = \ln (5)$

Step 5

Divide each term in

$12 t \ln (1.005) = \ln (5)$ by

$12 \ln (1.005)$ and simplify.

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Step 5.1

Divide each term in

$12 t \ln (1.005) = \ln (5)$ by

$12 \ln (1.005)$ .

$\frac{12 t \ln (1.005)}{12 \ln (1.005)} = \frac{\ln (5)}{12 \ln (1.005)}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of

$12$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{12 t \ln (1.005)}{12 \ln (1.005)} = \frac{\ln (5)}{12 \ln (1.005)}$

Step 5.2.1.2

Rewrite the expression.

$\frac{t \ln (1.005)}{\ln (1.005)} = \frac{\ln (5)}{12 \ln (1.005)}$

Step 5.2.2

Cancel the common factor of

$\ln (1.005)$ .

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Step 5.2.2.1

Cancel the common factor.

$\frac{t \ln (1.005)}{\ln (1.005)} = \frac{\ln (5)}{12 \ln (1.005)}$

Step 5.2.2.2

Divide

$t$ by

$1$ .

$t = \frac{\ln (5)}{12 \ln (1.005)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$t = \frac{\ln (5)}{12 \ln (1.005)}$

Decimal Form:

$t = 26.89096937 \dots$