Trigonometry Examples

$10000 = 22500 \cdot 10^{- \frac{t}{12}}$

Step 1

Rewrite the equation as $22500 \cdot 10^{- \frac{t}{12}} = 10000$ .

$22500 \cdot 10^{- \frac{t}{12}} = 10000$

Step 2

Divide each term in $22500 \cdot 10^{- \frac{t}{12}} = 10000$ by $22500$ and simplify.

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Step 2.1

Divide each term in $22500 \cdot 10^{- \frac{t}{12}} = 10000$ by $22500$ .

$\frac{22500 \cdot 10^{- \frac{t}{12}}}{22500} = \frac{10000}{22500}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $22500$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{22500 \cdot 10^{- \frac{t}{12}}}{22500} = \frac{10000}{22500}$

Step 2.2.1.2

Divide $10^{- \frac{t}{12}}$ by $1$ .

$10^{- \frac{t}{12}} = \frac{10000}{22500}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $10000$ and $22500$ .

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Step 2.3.1.1

Factor $2500$ out of $10000$ .

$10^{- \frac{t}{12}} = \frac{2500 (4)}{22500}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $2500$ out of $22500$ .

$10^{- \frac{t}{12}} = \frac{2500 \cdot 4}{2500 \cdot 9}$

Step 2.3.1.2.2

Cancel the common factor.

$10^{- \frac{t}{12}} = \frac{2500 \cdot 4}{2500 \cdot 9}$

Step 2.3.1.2.3

Rewrite the expression.

$10^{- \frac{t}{12}} = \frac{4}{9}$

Step 3

Take the base $10$ logarithm of both sides of the equation to remove the variable from the exponent.

$\log (10^{- \frac{t}{12}}) = \log (\frac{4}{9})$

Step 4

Expand the left side.

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Step 4.1

Expand $\log (10^{- \frac{t}{12}})$ by moving $- \frac{t}{12}$ outside the logarithm.

$- \frac{t}{12} \log (10) = \log (\frac{4}{9})$

Step 4.2

Logarithm base $10$ of $10$ is $1$ .

$- \frac{t}{12} \cdot 1 = \log (\frac{4}{9})$

Step 4.3

Multiply $- 1$ by $1$ .

$- \frac{t}{12} = \log (\frac{4}{9})$

Step 5

Multiply both sides of the equation by $- 12$ .

$- 12 (- \frac{t}{12}) = - 12 \log (\frac{4}{9})$

Step 6

Simplify the left side.

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Step 6.1

Simplify $- 12 (- \frac{t}{12})$ .

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Step 6.1.1

Cancel the common factor of $12$ .

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Step 6.1.1.1

Move the leading negative in $- \frac{t}{12}$ into the numerator.

$- 12 \frac{- t}{12} = - 12 \log (\frac{4}{9})$

Step 6.1.1.2

Factor $12$ out of $- 12$ .

$12 (- 1) \frac{- t}{12} = - 12 \log (\frac{4}{9})$

Step 6.1.1.3

Cancel the common factor.

$12 \cdot - 1 \frac{- t}{12} = - 12 \log (\frac{4}{9})$

Step 6.1.1.4

Rewrite the expression.

$- - t = - 12 \log (\frac{4}{9})$

Step 6.1.2

Multiply.

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Step 6.1.2.1

Multiply $- 1$ by $- 1$ .

$1 t = - 12 \log (\frac{4}{9})$

Step 6.1.2.2

Multiply $t$ by $1$ .

$t = - 12 \log (\frac{4}{9})$

Step 7

The result can be shown in multiple forms.

Exact Form:

$t = - 12 \log (\frac{4}{9})$

Decimal Form:

$t = 4.22619021 \dots$