Growth vs Decay Math Example 4

Follow the full solution, then compare it with the other examples linked below.

hard

Show algebraically that

f(x)=e^{-0.3x}

is an exponential decay function by rewriting it in the form

a\cdot b^x

and confirming

0<b<1

1
Rewrite: $e^{-0.3x} = (e^{-0.3})^x$ . So $a=1$ and $b=e^{-0.3}$ .
2
Compute: $b=e^{-0.3}\approx0.7408$ . Since $0<0.7408<1$ , this is exponential decay. ✓

Answer

f(x)=(e^{-0.3})^x\approx(0.7408)^x

; decay since

b\approx0.7408\in(0,1)

Negative exponents in

e^{-kx}

(with

k>0

) always produce decay. Rewriting as

(e^{-k})^x

reveals the base, which is always between

0

and

1

when

k>0

About Growth vs Decay

Exponential growth occurs when a quantity multiplies by a factor $> 1$ repeatedly; exponential decay when it multiplies by a factor between 0 and 1.

Classify each function as growth or decay, and find its value at [formula]: (a) [formula], (b) [form

A radioactive substance has a half-life of [formula] years. Starting with [formula] g, write the dec

A population grows by [formula] per year. Starting at [formula], write the growth function and find