Question

Consider the curve y=f(x)=(2^x)-1

Find a value for a such that the average value of the function f(x) on the interval [0,a] is equal to 1.

Answer 1

(1/a-0) ∫ [0,a] (2^x-1)dx = 1
I do not know how to solve for a

Answer 2

did you apply the value a

Answer 3

I found the antiderivative and I applied the first fundamental theorem of calculus. I'm getting stuck with simplification.

1/a[(2^a/ln(2))-a)-(2^0/ln(2))]

Answer 4

ok

Answer 5

I've gotten as far as:
2^a-aln(2)-1/aln(2)=1

Answer 6

that looks right so far

Answer 7

2^a-1=-2aln(2)

Answer 8

we can really only approximate it

Answer 9

a=1.812647074

Answer 10

Yeah, that's what I got. So there's easy no way to simplify?

Answer 11

no, not really.

Answer 12

unless you want to leave your answer in terms of the Lambert function

Answer 13

No, I think this is good. Thanks for the help.