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A virus takes $6$ days to double its original population $\left(A=2A_0\right)$. How long will it take to triple its population?

A bacteria doubles its original population in $24$hours $\left(A=2A_0\right)$. How big will its population be in $72$ hours?

Carbon-$14$is used for archeological carbon dating Its half-life is $5,730$years. How much of a 100 -gram sample of Carbon-$14$ will be left in $1000$ years?

Radioactive technetium-$99$m is often used in diagnostic medicine as it has a relatively short half-life but lasts long enough to get the needed testing done on the patient. If its half-life is $6$hours, how much of the radioactive material form a $0.5\mathrm{ml}$ injection will be in the body in $24$ hours?

Explain the method you would use to solve these equations: $3^{x+1}=81,3^{x+1}=75$. Does your method require logarithms for both equations? Why or why not?

What is the difference between the equation for exponential growth versus the equation for exponential decay?

In the following exercises, for each pair of functions, find $\left(a\right)\left(f∘g\right)\left(x\right),\left(b\right)\left(g∘f\right)\left(x\right),\left(c\right)\left(f·g\right)\left(x\right)$.

$f\left(x\right)=7x-2$and

$g\left(x\right)=5x+1$

In the following exercises, for each pair of functions, find $\left(a\right)\left(f∘g\right)\left(x\right),\left(b\right)\left(g∘f\right)\left(x\right),\left(c\right)\left(f·g\right)\left(x\right)$

$f\left(x\right)=4x$and

$g\left(x\right)=x^2+3x$

For functions

$f\left(x\right)=3x^2+2$and

$g\left(x\right)=4x-3$, find

localid="1645087490490" $$\begin{gathered} \left(a\right)\left(f∘g\right)(-3) \\ \left(b\right)\left(g∘f\right)(-2) \\ \left(c\right)\left(f∘f\right)(-1) \end{gathered}$$

For functions

$f\left(x\right)=2x^3+5$and

$g\left(x\right)=3x^2-7$,find

$$\begin{gathered} \left(a\right)\left(f∘g\right)(-1) \\ \left(b\right)\left(g∘f\right)(-2) \\ \left(c\right)\left(g∘g\right)\left(1\right) \end{gathered}$$