# Math - Rates of change and the number e (check + h

Let f(x)= (1/b)^x for b>1, and left g(x) = (f'(x))/(f(x))

a) Predict the shape of the graph of g(x).

it will be a straight line, since ln(1/b) will be a constant

b) Test your prediction by exploring two specific cases.

Values of (1/2) and (1/10)

c) Summarize your findings, using words and diagrams.

repeat part a) expect in words and graphs

d) For what value of b will g(x)= -1

it would be undefined, since there is no value of b, and you prove it by taking natural log on both sides and showing that ln -1 is a non real answer

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1. (a) Correct.
Also, note that since the function is a constant, it does not contain x as a variable, therefore it is a horizontal line y=-log(b).
(d)For what value of b will g(x)= -1
g(x)=-ln(b)
so
-ln(b)=-1
ln(b)=1
raise to the power of e:
e^(ln(b)) = e^1
b=e (=2.718...)

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