Precalculus Examples

$0 = 96.2 {(\frac{1}{2})}^{\frac{t}{14.3}} - 12.5$

Step 1

Rewrite the equation as $96.2 {(\frac{1}{2})}^{\frac{t}{14.3}} - 12.5 = 0$ .

$96.2 {(\frac{1}{2})}^{\frac{t}{14.3}} - 12.5 = 0$

Step 2

Simplify each term.

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Step 2.1

Multiply by $1$ .

$96.2 {(\frac{1}{2})}^{\frac{1 t}{14.3}} - 12.5 = 0$

Step 2.2

Factor $14.3$ out of $14.3$ .

$96.2 {(\frac{1}{2})}^{\frac{1 t}{14.3 (1)}} - 12.5 = 0$

Step 2.3

Separate fractions.

$96.2 {(\frac{1}{2})}^{\frac{1}{14.3} \cdot \frac{t}{1}} - 12.5 = 0$

Step 2.4

Divide $1$ by $14.3$ .

$96.2 {(\frac{1}{2})}^{0.\overline{069930} \frac{t}{1}} - 12.5 = 0$

Step 2.5

Divide $t$ by $1$ .

$96.2 {(\frac{1}{2})}^{0.\overline{069930} t} - 12.5 = 0$

Step 2.6

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

$96.2 \frac{1^{0.\overline{069930} t}}{2^{0.\overline{069930} t}} - 12.5 = 0$

Step 2.7

One to any power is one.

$96.2 \frac{1}{2^{0.\overline{069930} t}} - 12.5 = 0$

Step 2.8

Combine $96.2$ and $\frac{1}{2^{0.\overline{069930} t}}$ .

$\frac{96.2}{2^{0.\overline{069930} t}} - 12.5 = 0$

Step 3

Add $12.5$ to both sides of the equation.

$\frac{96.2}{2^{0.\overline{069930} t}} = 12.5$

Step 4

Multiply both sides by $2^{0.\overline{069930} t}$ .

$\frac{96.2}{2^{0.\overline{069930} t}} \cdot 2^{0.\overline{069930} t} = 12.5 \cdot 2^{0.\overline{069930} t}$

Step 5

Simplify the left side.

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Step 5.1

Cancel the common factor of $2^{0.\overline{069930} t}$ .

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Step 5.1.1

Cancel the common factor.

$\frac{96.2}{2^{0.\overline{069930} t}} \cdot 2^{0.\overline{069930} t} = 12.5 \cdot 2^{0.\overline{069930} t}$

Step 5.1.2

Rewrite the expression.

$96.2 = 12.5 \cdot 2^{0.\overline{069930} t}$

Step 6

Solve for $t$ .

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Step 6.1

Rewrite the equation as $12.5 \cdot 2^{0.\overline{069930} t} = 96.2$ .

$12.5 \cdot 2^{0.\overline{069930} t} = 96.2$

Step 6.2

Divide each term in $12.5 \cdot 2^{0.\overline{069930} t} = 96.2$ by $12.5$ and simplify.

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Step 6.2.1

Divide each term in $12.5 \cdot 2^{0.\overline{069930} t} = 96.2$ by $12.5$ .

$\frac{12.5 \cdot 2^{0.\overline{069930} t}}{12.5} = \frac{96.2}{12.5}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $12.5$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{12.5 \cdot 2^{0.\overline{069930} t}}{12.5} = \frac{96.2}{12.5}$

Step 6.2.2.1.2

Divide $2^{0.\overline{069930} t}$ by $1$ .

$2^{0.\overline{069930} t} = \frac{96.2}{12.5}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Divide $96.2$ by $12.5$ .

$2^{0.\overline{069930} t} = 7.696$

Step 6.3

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

$\ln (2^{0.\overline{069930} t}) = \ln (7.696)$

Step 6.4

Expand the left side.

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Step 6.4.1

Expand $\ln (2^{0.\overline{069930} t})$ by moving $0.\overline{069930} t$ outside the logarithm.

$0.\overline{069930} t \ln (2) = \ln (7.696)$

Step 6.4.2

Multiply $\ln (2)$ by $0.\overline{069930}$ .

$0.04847183 t = \ln (7.696)$

Step 6.5

Divide each term in $0.04847183 t = \ln (7.696)$ by $0.04847183$ and simplify.

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Step 6.5.1

Divide each term in $0.04847183 t = \ln (7.696)$ by $0.04847183$ .

$\frac{0.04847183 t}{0.04847183} = \frac{\ln (7.696)}{0.04847183}$

Step 6.5.2

Simplify the left side.

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Step 6.5.2.1

Cancel the common factor of $0.04847183$ .

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Step 6.5.2.1.1

Cancel the common factor.

$\frac{0.04847183 t}{0.04847183} = \frac{\ln (7.696)}{0.04847183}$

Step 6.5.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (7.696)}{0.04847183}$

Step 6.5.3

Simplify the right side.

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Step 6.5.3.1

Replace $e$ with an approximation.

$t = \frac{\log_{2.71828182} (7.696)}{0.04847183}$

Step 6.5.3.2

Log base $2.71828182$ of $7.696$ is approximately $2.04070071$ .

$t = \frac{2.04070071}{0.04847183}$

Step 6.5.3.3

Divide $2.04070071$ by $0.04847183$ .

$t = 42.10075582$