Expert: Paul Klarreich - 1/17/2007

Question
Our text makes it clear that for y = a b^x if a>0 and b>1 then the equation represents exponential growth. If a>0 and 0<b>1 then we have exponential decay. However, even though graphs are shown where a<0 it is not clear whether the equations are reclassified as decay and growth or not. (i.e. in y = ab^x with a<0 and b>0 is this decay? and if a<0 while 0<b<1 is this growth?) Can you clarify this for us?

Answer
Questioner:   Michael
Category:  Advanced Math
 
Subject:  exponential equations
Question:  Our text makes it clear that for y = a b^x if a>0 and b>1 then the equation represents exponential growth. If a>0 and 0<b>1 then we have exponential decay.

However, even though graphs are shown where a<0 it is not clear whether the equations are reclassified as decay and growth or not. (i.e. in y = ab^x with a<0 and b>0 is this decay? and if a<0 while 0<b<1 is this growth?) Can you clarify this for us?
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Hi, Michael,

The usual difficulty in situations like this is the failure to clearly define terms.  

Ordinarily, when you have the 'growth and decay' equation:

y = a b^x,

it is in the form:

N = N0 e^(lambda x)

and N0, the number of (bacteria, atoms of radioactive substance, whatever) at t=0, is positive, because the application doesn't make sense otherwise.

But the equation can certainly have a negative value of a.  So, I make this suggestion for clarifying definitions:

Growth always means that the value of y goes away from zero.
Decay  always means that the value of y goes towards   zero.

In that case, when b < 1, the absolute value of y decreases, so y is approaching zero, and it is decay.

Likewise, if  b > 1, | y | is increasing, so y is moving away from zero, and we have growth.