Algebra Examples

$20 = 100 {(\frac{1}{2})}^{\frac{t}{214}}$

Step 1

Rewrite the equation as $100 {(\frac{1}{2})}^{\frac{t}{214}} = 20$ .

$100 {(\frac{1}{2})}^{\frac{t}{214}} = 20$

Step 2

Divide each term in $100 {(\frac{1}{2})}^{\frac{t}{214}} = 20$ by $100$ and simplify.

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Step 2.1

Divide each term in $100 {(\frac{1}{2})}^{\frac{t}{214}} = 20$ by $100$ .

$\frac{100 {(\frac{1}{2})}^{\frac{t}{214}}}{100} = \frac{20}{100}$

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.

$\frac{100 {(\frac{1}{2})}^{\frac{t}{214}}}{100} = \frac{20}{100}$

Step 2.2.1.2

Divide ${(\frac{1}{2})}^{\frac{t}{214}}$ by $1$ .

${(\frac{1}{2})}^{\frac{t}{214}} = \frac{20}{100}$

Step 2.2.2

Simplify the expression.

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Step 2.2.2.1

Apply the product rule to $\frac{1}{2}$ .

$\frac{1^{\frac{t}{214}}}{2^{\frac{t}{214}}} = \frac{20}{100}$

Step 2.2.2.2

One to any power is one.

$\frac{1}{2^{\frac{t}{214}}} = \frac{20}{100}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $20$ and $100$ .

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Step 2.3.1.1

Factor $20$ out of $20$ .

$\frac{1}{2^{\frac{t}{214}}} = \frac{20 (1)}{100}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $20$ out of $100$ .

$\frac{1}{2^{\frac{t}{214}}} = \frac{20 \cdot 1}{20 \cdot 5}$

Step 2.3.1.2.2

Cancel the common factor.

$\frac{1}{2^{\frac{t}{214}}} = \frac{20 \cdot 1}{20 \cdot 5}$

Step 2.3.1.2.3

Rewrite the expression.

$\frac{1}{2^{\frac{t}{214}}} = \frac{1}{5}$

Step 3

Multiply both sides by $2^{\frac{t}{214}}$ .

$\frac{1}{2^{\frac{t}{214}}} \cdot 2^{\frac{t}{214}} = \frac{1}{5} \cdot 2^{\frac{t}{214}}$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $2^{\frac{t}{214}}$ .

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Step 4.1.1.1

Cancel the common factor.

$\frac{1}{2^{\frac{t}{214}}} \cdot 2^{\frac{t}{214}} = \frac{1}{5} \cdot 2^{\frac{t}{214}}$

Step 4.1.1.2

Rewrite the expression.

$1 = \frac{1}{5} \cdot 2^{\frac{t}{214}}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Combine $\frac{1}{5}$ and $2^{\frac{t}{214}}$ .

$1 = \frac{2^{\frac{t}{214}}}{5}$

Step 5

Solve for $t$ .

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Step 5.1

Rewrite the equation as $\frac{2^{\frac{t}{214}}}{5} = 1$ .

$\frac{2^{\frac{t}{214}}}{5} = 1$

Step 5.2

Multiply both sides of the equation by $5$ .

$5 \frac{2^{\frac{t}{214}}}{5} = 5 \cdot 1$

Step 5.3

Simplify both sides of the equation.

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Step 5.3.1

Simplify the left side.

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Step 5.3.1.1

Cancel the common factor of $5$ .

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Step 5.3.1.1.1

Cancel the common factor.

$5 \frac{2^{\frac{t}{214}}}{5} = 5 \cdot 1$

Step 5.3.1.1.2

Rewrite the expression.

$2^{\frac{t}{214}} = 5 \cdot 1$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Multiply $5$ by $1$ .

$2^{\frac{t}{214}} = 5$

Step 5.4

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

$\ln (2^{\frac{t}{214}}) = \ln (5)$

Step 5.5

Expand the left side.

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Step 5.5.1

Expand $\ln (2^{\frac{t}{214}})$ by moving $\frac{t}{214}$ outside the logarithm.

$\frac{t}{214} \ln (2) = \ln (5)$

Step 5.5.2

Combine $\frac{t}{214}$ and $\ln (2)$ .

$\frac{t \ln (2)}{214} = \ln (5)$

Step 5.6

Multiply both sides of the equation by $\frac{214}{\ln (2)}$ .

$\frac{214}{\ln (2)} \cdot \frac{t \ln (2)}{214} = \frac{214}{\ln (2)} \ln (5)$

Step 5.7

Simplify both sides of the equation.

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Step 5.7.1

Simplify the left side.

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Step 5.7.1.1

Simplify $\frac{214}{\ln (2)} \cdot \frac{t \ln (2)}{214}$ .

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Step 5.7.1.1.1

Cancel the common factor of $214$ .

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Step 5.7.1.1.1.1

Cancel the common factor.

$\frac{214}{\ln (2)} \cdot \frac{t \ln (2)}{214} = \frac{214}{\ln (2)} \ln (5)$

Step 5.7.1.1.1.2

Rewrite the expression.

$\frac{1}{\ln (2)} (t \ln (2)) = \frac{214}{\ln (2)} \ln (5)$

Step 5.7.1.1.2

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

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Step 5.7.1.1.2.1

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

$\frac{1}{\ln (2)} (\ln (2) t) = \frac{214}{\ln (2)} \ln (5)$

Step 5.7.1.1.2.2

Cancel the common factor.

$\frac{1}{\ln (2)} (\ln (2) t) = \frac{214}{\ln (2)} \ln (5)$

Step 5.7.1.1.2.3

Rewrite the expression.

$t = \frac{214}{\ln (2)} \ln (5)$

Step 5.7.2

Simplify the right side.

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Step 5.7.2.1

Combine $\frac{214}{\ln (2)}$ and $\ln (5)$ .

$t = \frac{214 \ln (5)}{\ln (2)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$t = \frac{214 \ln (5)}{\ln (2)}$

Decimal Form:

$t = 496.89261230 \dots$