$\"\"$

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The exponential function is $\"\"$ .

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Half life of the isotope is $\"\"$ .

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At half life time, $\"\"$ .

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$\"\"$

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$\"\"$

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Apply logarithm on each side.

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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Therefore, the exponential equation is $\"\"$ .

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$\"\"$

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The amount of isotope after $\"\"$ years is $\"\"$ .

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Find $\"\"$ .

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Substitute $\"\"$ and $\"\"$ in the exponential equation.

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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Initial quantity is $\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$ .

\

$\"\"$

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Find the amount of isotope after $\"\"$ years.

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Substitute $\"\"$ in the exponential equation.

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

The amount of isotope after $\"\"$ years is $\"\"$ .

\

$\"\"$

\

Initial quantity is $\"\"$ .

\

The amount of isotope after $\"\"$ years is $\"\"$ .