Precalculus Examples

$9000 = 12000 \cdot 2^{- \frac{t}{15}}$

Step 1

Rewrite the equation as $12000 \cdot 2^{- \frac{t}{15}} = 9000$ .

$12000 \cdot 2^{- \frac{t}{15}} = 9000$

Step 2

Divide each term in $12000 \cdot 2^{- \frac{t}{15}} = 9000$ by $12000$ and simplify.

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Step 2.1

Divide each term in $12000 \cdot 2^{- \frac{t}{15}} = 9000$ by $12000$ .

$\frac{12000 \cdot 2^{- \frac{t}{15}}}{12000} = \frac{9000}{12000}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $12000$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{12000 \cdot 2^{- \frac{t}{15}}}{12000} = \frac{9000}{12000}$

Step 2.2.1.2

Divide $2^{- \frac{t}{15}}$ by $1$ .

$2^{- \frac{t}{15}} = \frac{9000}{12000}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $9000$ and $12000$ .

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Step 2.3.1.1

Factor $3000$ out of $9000$ .

$2^{- \frac{t}{15}} = \frac{3000 (3)}{12000}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $3000$ out of $12000$ .

$2^{- \frac{t}{15}} = \frac{3000 \cdot 3}{3000 \cdot 4}$

Step 2.3.1.2.2

Cancel the common factor.

$2^{- \frac{t}{15}} = \frac{3000 \cdot 3}{3000 \cdot 4}$

Step 2.3.1.2.3

Rewrite the expression.

$2^{- \frac{t}{15}} = \frac{3}{4}$

Step 3

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

$\ln (2^{- \frac{t}{15}}) = \ln (\frac{3}{4})$

Step 4

Expand the left side.

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Step 4.1

Expand $\ln (2^{- \frac{t}{15}})$ by moving $- \frac{t}{15}$ outside the logarithm.

$- \frac{t}{15} \ln (2) = \ln (\frac{3}{4})$

Step 4.2

Combine $\ln (2)$ and $\frac{t}{15}$ .

$- \frac{\ln (2) t}{15} = \ln (\frac{3}{4})$

Step 5

Simplify the left side.

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Step 5.1

Reorder factors in $- \frac{\ln (2) t}{15}$ .

$- \frac{t \ln (2)}{15} = \ln (\frac{3}{4})$

Step 6

Multiply both sides of the equation by $- \frac{15}{\ln (2)}$ .

$- \frac{15}{\ln (2)} (- \frac{t \ln (2)}{15}) = - \frac{15}{\ln (2)} \ln (\frac{3}{4})$

Step 7

Simplify both sides of the equation.

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Step 7.1

Simplify the left side.

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Step 7.1.1

Simplify $- \frac{15}{\ln (2)} (- \frac{t \ln (2)}{15})$ .

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Step 7.1.1.1

Cancel the common factor of $15$ .

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Step 7.1.1.1.1

Move the leading negative in $- \frac{15}{\ln (2)}$ into the numerator.

$\frac{- 15}{\ln (2)} (- \frac{t \ln (2)}{15}) = - \frac{15}{\ln (2)} \ln (\frac{3}{4})$

Step 7.1.1.1.2

Move the leading negative in $- \frac{t \ln (2)}{15}$ into the numerator.

$\frac{- 15}{\ln (2)} \cdot \frac{- t \ln (2)}{15} = - \frac{15}{\ln (2)} \ln (\frac{3}{4})$

Step 7.1.1.1.3

Factor $15$ out of $- 15$ .

$\frac{15 (- 1)}{\ln (2)} \cdot \frac{- t \ln (2)}{15} = - \frac{15}{\ln (2)} \ln (\frac{3}{4})$

Step 7.1.1.1.4

Cancel the common factor.

$\frac{15 \cdot - 1}{\ln (2)} \cdot \frac{- t \ln (2)}{15} = - \frac{15}{\ln (2)} \ln (\frac{3}{4})$

Step 7.1.1.1.5

Rewrite the expression.

$\frac{- 1}{\ln (2)} (- t \ln (2)) = - \frac{15}{\ln (2)} \ln (\frac{3}{4})$

Step 7.1.1.2

Cancel the common factor of $\ln (2)$ .

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Step 7.1.1.2.1

Factor $\ln (2)$ out of $- t \ln (2)$ .

$\frac{- 1}{\ln (2)} (\ln (2) (- t)) = - \frac{15}{\ln (2)} \ln (\frac{3}{4})$

Step 7.1.1.2.2

Cancel the common factor.

$\frac{- 1}{\ln (2)} (\ln (2) (- t)) = - \frac{15}{\ln (2)} \ln (\frac{3}{4})$

Step 7.1.1.2.3

Rewrite the expression.

$- - t = - \frac{15}{\ln (2)} \ln (\frac{3}{4})$

Step 7.1.1.3

Multiply.

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Step 7.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 t = - \frac{15}{\ln (2)} \ln (\frac{3}{4})$

Step 7.1.1.3.2

Multiply $t$ by $1$ .

$t = - \frac{15}{\ln (2)} \ln (\frac{3}{4})$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $- \frac{15}{\ln (2)} \ln (\frac{3}{4})$ .

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Step 7.2.1.1

Combine $\ln (\frac{3}{4})$ and $\frac{15}{\ln (2)}$ .

$t = - \frac{\ln (\frac{3}{4}) \cdot 15}{\ln (2)}$

Step 7.2.1.2

Move $15$ to the left of $\ln (\frac{3}{4})$ .

$t = - \frac{15 \ln (\frac{3}{4})}{\ln (2)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$t = - \frac{15 \ln (\frac{3}{4})}{\ln (2)}$

Decimal Form:

$t = 6.22556248 \dots$