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Again, we have the form $$y=A_0e^$$ where $$A_0$$ is the starting value, and $$e$$ is Euler’s constant.

Now $$k$$ is a negative constant that determines the rate of decay.

To describe these numbers, we often use orders of magnitude.

The order of magnitude is the power of ten, when the number is expressed in scientific notation, with one digit to the left of the decimal.

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We have already explored some basic applications of exponential and logarithmic functions.

This gives us the half-life formula $t=−\dfrac$ Example $$\Page Index$$: Finding the Function that Describes Radioactive Decay The half-life of carbon-14 is $$5,730$$ years.Express the amount of carbon-14 remaining as a function of time, $$t$$. $\begin A&= A_0e^ \qquad \text\ 0.5A_0&= A_0e^ \qquad \text 0.5A_0 \text f(t)\ 0.5&= e^ \qquad \text A_0\ \ln(0.5)&= 5730k \qquad \text\ k&= \dfrac \qquad \text\ A&= A_0e^ \qquad \text \end$ The function that describes this continuous decay is $$f(t)=A_0e^$$.We observe that the coefficient of $$t$$, $$\dfrac≈−1.2097×10^$$ is negative, as expected in the case of exponential decay.It compares the difference between the ratio of two isotopes of carbon in an organic artifact or fossil to the ratio of those two isotopes in the air.It is believed to be accurate to within about $$1\%$$ error for plants or animals that died within the last $$60,000$$ years.