Finite Math Examples

$1000 {(1.08)}^{2 t} = 5$ , $0$

Step 1

Divide each term in $1000 \cdot {1.08}^{2 t} = 5$ by $1000$ and simplify.

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Step 1.1

Divide each term in $1000 \cdot {1.08}^{2 t} = 5$ by $1000$ .

$\frac{1000 \cdot {1.08}^{2 t}}{1000} = \frac{5}{1000}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $1000$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{1000 \cdot {1.08}^{2 t}}{1000} = \frac{5}{1000}$

Step 1.2.1.2

Divide ${1.08}^{2 t}$ by $1$ .

${1.08}^{2 t} = \frac{5}{1000}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Cancel the common factor of $5$ and $1000$ .

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Step 1.3.1.1

Factor $5$ out of $5$ .

${1.08}^{2 t} = \frac{5 (1)}{1000}$

Step 1.3.1.2

Cancel the common factors.

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Step 1.3.1.2.1

Factor $5$ out of $1000$ .

${1.08}^{2 t} = \frac{5 \cdot 1}{5 \cdot 200}$

Step 1.3.1.2.2

Cancel the common factor.

${1.08}^{2 t} = \frac{5 \cdot 1}{5 \cdot 200}$

Step 1.3.1.2.3

Rewrite the expression.

${1.08}^{2 t} = \frac{1}{200}$

Step 2

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

$\ln ({1.08}^{2 t}) = \ln (\frac{1}{200})$

Step 3

Expand $\ln ({1.08}^{2 t})$ by moving $2 t$ outside the logarithm.

$2 t \ln (1.08) = \ln (\frac{1}{200})$

Step 4

Divide each term in $2 t \ln (1.08) = \ln (\frac{1}{200})$ by $2 \ln (1.08)$ and simplify.

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Step 4.1

Divide each term in $2 t \ln (1.08) = \ln (\frac{1}{200})$ by $2 \ln (1.08)$ .

$\frac{2 t \ln (1.08)}{2 \ln (1.08)} = \frac{\ln (\frac{1}{200})}{2 \ln (1.08)}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 t \ln (1.08)}{2 \ln (1.08)} = \frac{\ln (\frac{1}{200})}{2 \ln (1.08)}$

Step 4.2.1.2

Rewrite the expression.

$\frac{t \ln (1.08)}{\ln (1.08)} = \frac{\ln (\frac{1}{200})}{2 \ln (1.08)}$

Step 4.2.2

Cancel the common factor of $\ln (1.08)$ .

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Step 4.2.2.1

Cancel the common factor.

$\frac{t \ln (1.08)}{\ln (1.08)} = \frac{\ln (\frac{1}{200})}{2 \ln (1.08)}$

Step 4.2.2.2

Divide $t$ by $1$ .

$t = \frac{\ln (\frac{1}{200})}{2 \ln (1.08)}$

Step 5

Replace the variable $t$ with $\frac{\ln (\frac{1}{200})}{2 \ln (1.08)}$ in the expression.

$0$