Precalculus Examples

$10000 = 100 \cdot 2^{\frac{t}{10}}$

Step 1

Rewrite the equation as $100 \cdot 2^{\frac{t}{10}} = 10000$ .

$100 \cdot 2^{\frac{t}{10}} = 10000$

Step 2

Divide each term in $100 \cdot 2^{\frac{t}{10}} = 10000$ by $100$ and simplify.

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Step 2.1

Divide each term in $100 \cdot 2^{\frac{t}{10}} = 10000$ by $100$ .

$\frac{100 \cdot 2^{\frac{t}{10}}}{100} = \frac{10000}{100}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $100$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{100 \cdot 2^{\frac{t}{10}}}{100} = \frac{10000}{100}$

Step 2.2.1.2

Divide $2^{\frac{t}{10}}$ by $1$ .

$2^{\frac{t}{10}} = \frac{10000}{100}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $10000$ by $100$ .

$2^{\frac{t}{10}} = 100$

Step 3

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

$\ln (2^{\frac{t}{10}}) = \ln (100)$

Step 4

Expand the left side.

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Step 4.1

Expand $\ln (2^{\frac{t}{10}})$ by moving $\frac{t}{10}$ outside the logarithm.

$\frac{t}{10} \ln (2) = \ln (100)$

Step 4.2

Combine $\frac{t}{10}$ and $\ln (2)$ .

$\frac{t \ln (2)}{10} = \ln (100)$

Step 5

Multiply both sides of the equation by $\frac{10}{\ln (2)}$ .

$\frac{10}{\ln (2)} \cdot \frac{t \ln (2)}{10} = \frac{10}{\ln (2)} \ln (100)$

Step 6

Simplify both sides of the equation.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Simplify $\frac{10}{\ln (2)} \cdot \frac{t \ln (2)}{10}$ .

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Step 6.1.1.1

Cancel the common factor of $10$ .

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Step 6.1.1.1.1

Cancel the common factor.

$\frac{10}{\ln (2)} \cdot \frac{t \ln (2)}{10} = \frac{10}{\ln (2)} \ln (100)$

Step 6.1.1.1.2

Rewrite the expression.

$\frac{1}{\ln (2)} (t \ln (2)) = \frac{10}{\ln (2)} \ln (100)$

Step 6.1.1.2

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

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Step 6.1.1.2.1

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

$\frac{1}{\ln (2)} (\ln (2) t) = \frac{10}{\ln (2)} \ln (100)$

Step 6.1.1.2.2

Cancel the common factor.

$\frac{1}{\ln (2)} (\ln (2) t) = \frac{10}{\ln (2)} \ln (100)$

Step 6.1.1.2.3

Rewrite the expression.

$t = \frac{10}{\ln (2)} \ln (100)$

Step 6.2

Simplify the right side.

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Step 6.2.1

Combine $\frac{10}{\ln (2)}$ and $\ln (100)$ .

$t = \frac{10 \ln (100)}{\ln (2)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$t = \frac{10 \ln (100)}{\ln (2)}$

Decimal Form:

$t = 66.43856189 \dots$