Precalculus Examples

$\frac{1}{2} {(10)}^{3 t} = 100$

Step 1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} \cdot 10^{3 t}) = 2 \cdot 100$

Step 2

Simplify both sides of the equation.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $2 (\frac{1}{2} \cdot 10^{3 t})$ .

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Step 2.1.1.1

Combine $\frac{1}{2}$ and $10^{3 t}$ .

$2 \frac{10^{3 t}}{2} = 2 \cdot 100$

Step 2.1.1.2

Cancel the common factor of $2$ .

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Step 2.1.1.2.1

Cancel the common factor.

$2 \frac{10^{3 t}}{2} = 2 \cdot 100$

Step 2.1.1.2.2

Rewrite the expression.

$10^{3 t} = 2 \cdot 100$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $2$ by $100$ .

$10^{3 t} = 200$

Step 3

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

$\log (10^{3 t}) = \log (200)$

Step 4

Expand the left side.

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Step 4.1

Expand $\log (10^{3 t})$ by moving $3 t$ outside the logarithm.

$3 t \log (10) = \log (200)$

Step 4.2

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

$3 t \cdot 1 = \log (200)$

Step 4.3

Multiply $3$ by $1$ .

$3 t = \log (200)$

Step 5

Divide each term in $3 t = \log (200)$ by $3$ and simplify.

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Step 5.1

Divide each term in $3 t = \log (200)$ by $3$ .

$\frac{3 t}{3} = \frac{\log (200)}{3}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $3$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{3 t}{3} = \frac{\log (200)}{3}$

Step 5.2.1.2

Divide $t$ by $1$ .

$t = \frac{\log (200)}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$t = \frac{\log (200)}{3}$

Decimal Form:

$t = 0.76700999 \dots$