Finite Math Examples

$8000 e^{0.06 t} = 47000 {(1 + \frac{0.03}{1})}^{t}$

Step 1

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

$\ln (8000 e^{0.06 t}) = \ln (47000 {(1 + \frac{0.03}{1})}^{t})$

Step 2

Expand the left side.

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Step 2.1

Rewrite $\ln (8000 e^{0.06 t})$ as $\ln (8000) + \ln (e^{0.06 t})$ .

$\ln (8000) + \ln (e^{0.06 t}) = \ln (47000 {(1 + \frac{0.03}{1})}^{t})$

Step 2.2

Expand $\ln (e^{0.06 t})$ by moving $0.06 t$ outside the logarithm.

$\ln (8000) + 0.06 t \ln (e) = \ln (47000 {(1 + \frac{0.03}{1})}^{t})$

Step 2.3

The natural logarithm of $e$ is $1$ .

$\ln (8000) + 0.06 t \cdot 1 = \ln (47000 {(1 + \frac{0.03}{1})}^{t})$

Step 2.4

Multiply $0.06$ by $1$ .

$\ln (8000) + 0.06 t = \ln (47000 {(1 + \frac{0.03}{1})}^{t})$

Step 3

Expand the right side.

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Step 3.1

Rewrite $\ln (47000 {(1 + \frac{0.03}{1})}^{t})$ as $\ln (47000) + \ln ({(1 + \frac{0.03}{1})}^{t})$ .

$\ln (8000) + 0.06 t = \ln (47000) + \ln ({(1 + \frac{0.03}{1})}^{t})$

Step 3.2

Expand $\ln ({(1 + \frac{0.03}{1})}^{t})$ by moving $t$ outside the logarithm.

$\ln (8000) + 0.06 t = \ln (47000) + t \ln (1 + \frac{0.03}{1})$

Step 3.3

Divide $0.03$ by $1$ .

$\ln (8000) + 0.06 t = \ln (47000) + t \ln (1 + 0.03)$

Step 3.4

Add $1$ and $0.03$ .

$\ln (8000) + 0.06 t = \ln (47000) + t \ln (1.03)$

Step 4

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

$\ln (8000) - \ln (47000) - t \ln (1.03) = - 0.06 t$

Step 5

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

$\ln (\frac{8000}{47000}) - t \ln (1.03) = - 0.06 t$

Step 6

Cancel the common factor of $8000$ and $47000$ .

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Step 6.1

Factor $1000$ out of $8000$ .

$\ln (\frac{1000 (8)}{47000}) - t \ln (1.03) = - 0.06 t$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $1000$ out of $47000$ .

$\ln (\frac{1000 \cdot 8}{1000 \cdot 47}) - t \ln (1.03) = - 0.06 t$

Step 6.2.2

Cancel the common factor.

$\ln (\frac{1000 \cdot 8}{1000 \cdot 47}) - t \ln (1.03) = - 0.06 t$

Step 6.2.3

Rewrite the expression.

$\ln (\frac{8}{47}) - t \ln (1.03) = - 0.06 t$

Step 7

Add $0.06 t$ to both sides of the equation.

$\ln (\frac{8}{47}) - t \ln (1.03) + 0.06 t = 0$

Step 8

Subtract $\ln (\frac{8}{47})$ from both sides of the equation.

$- t \ln (1.03) + 0.06 t = - \ln (\frac{8}{47})$

Step 9

Factor $0.02 t$ out of $- t \ln (1.03) + 0.06 t$ .

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Step 9.1

Factor $0.02 t$ out of $- t \ln (1.03)$ .

$0.02 t (- 50 \ln (1.03)) + 0.06 t = - \ln (\frac{8}{47})$

Step 9.2

Factor $0.02 t$ out of $0.06 t$ .

$0.02 t (- 50 \ln (1.03)) + 0.02 t (3) = - \ln (\frac{8}{47})$

Step 9.3

Factor $0.02 t$ out of $0.02 t (- 50 \ln (1.03)) + 0.02 t (3)$ .

$0.02 t (- 50 \ln (1.03) + 3) = - \ln (\frac{8}{47})$

Step 10

Divide each term in $0.02 t (- 50 \ln (1.03) + 3) = - \ln (\frac{8}{47})$ by $0.02 (- 50 \ln (1.03) + 3)$ and simplify.

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Step 10.1

Divide each term in $0.02 t (- 50 \ln (1.03) + 3) = - \ln (\frac{8}{47})$ by $0.02 (- 50 \ln (1.03) + 3)$ .

$\frac{0.02 t (- 50 \ln (1.03) + 3)}{0.02 (- 50 \ln (1.03) + 3)} = \frac{- \ln (\frac{8}{47})}{0.02 (- 50 \ln (1.03) + 3)}$

Step 10.2

Simplify the left side.

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Step 10.2.1

Cancel the common factor of $0.02$ .

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Step 10.2.1.1

Cancel the common factor.

$\frac{0.02 t (- 50 \ln (1.03) + 3)}{0.02 (- 50 \ln (1.03) + 3)} = \frac{- \ln (\frac{8}{47})}{0.02 (- 50 \ln (1.03) + 3)}$

Step 10.2.1.2

Rewrite the expression.

$\frac{t (- 50 \ln (1.03) + 3)}{- 50 \ln (1.03) + 3} = \frac{- \ln (\frac{8}{47})}{0.02 (- 50 \ln (1.03) + 3)}$

Step 10.2.2

Cancel the common factor of $- 50 \ln (1.03) + 3$ .

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Step 10.2.2.1

Cancel the common factor.

$\frac{t (- 50 \ln (1.03) + 3)}{- 50 \ln (1.03) + 3} = \frac{- \ln (\frac{8}{47})}{0.02 (- 50 \ln (1.03) + 3)}$

Step 10.2.2.2

Divide $t$ by $1$ .

$t = \frac{- \ln (\frac{8}{47})}{0.02 (- 50 \ln (1.03) + 3)}$

Step 10.3

Simplify the right side.

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Step 10.3.1

Move the negative in front of the fraction.

$t = - \frac{\ln (\frac{8}{47})}{0.02 (- 50 \ln (1.03) + 3)}$

Step 10.3.2

Multiply by $1$ .

$t = - \frac{1 \ln (\frac{8}{47})}{0.02 (- 50 \ln (1.03) + 3)}$

Step 10.3.3

Separate fractions.

$t = - (\frac{1}{0.02} \cdot \frac{\ln (\frac{8}{47})}{- 50 \ln (1.03) + 3})$

Step 10.3.4

Divide $1$ by $0.02$ .

$t = - (50 \frac{\ln (\frac{8}{47})}{- 50 \ln (1.03) + 3})$

Step 10.3.5

Combine $50$ and $\frac{\ln (\frac{8}{47})}{- 50 \ln (1.03) + 3}$ .

$t = - \frac{50 \ln (\frac{8}{47})}{- 50 \ln (1.03) + 3}$

Step 10.3.6

Factor $- 1$ out of $- 50 \ln (1.03)$ .

$t = - \frac{50 \ln (\frac{8}{47})}{- (50 \ln (1.03)) + 3}$

Step 10.3.7

Rewrite $3$ as $- 1 (- 3)$ .

$t = - \frac{50 \ln (\frac{8}{47})}{- (50 \ln (1.03)) - 1 (- 3)}$

Step 10.3.8

Factor $- 1$ out of $- (50 \ln (1.03)) - 1 (- 3)$ .

$t = - \frac{50 \ln (\frac{8}{47})}{- (50 \ln (1.03) - 3)}$

Step 10.3.9

Simplify the expression.

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Step 10.3.9.1

Rewrite $- (50 \ln (1.03) - 3)$ as $- 1 (50 \ln (1.03) - 3)$ .

$t = - \frac{50 \ln (\frac{8}{47})}{- 1 (50 \ln (1.03) - 3)}$

Step 10.3.9.2

Move the negative in front of the fraction.

$t = - - \frac{50 \ln (\frac{8}{47})}{50 \ln (1.03) - 3}$

Step 10.3.9.3

Multiply $- 1$ by $- 1$ .

$t = 1 \frac{50 \ln (\frac{8}{47})}{50 \ln (1.03) - 3}$

Step 10.3.9.4

Multiply $\frac{50 \ln (\frac{8}{47})}{50 \ln (1.03) - 3}$ by $1$ .

$t = \frac{50 \ln (\frac{8}{47})}{50 \ln (1.03) - 3}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$t = \frac{50 \ln (\frac{8}{47})}{50 \ln (1.03) - 3}$

Decimal Form:

$t = 58.16808110 \dots$