What is the inverse of f(x)=2^x?

Answer 1

f(x)=2^x

let f(x)=y

then y=2^x
solve for x

ln y =ln 2^x
ln y = x ln 2
x=ln y/ ln 2

therefore...f^(-1)(x)=ln x/ln2

Answer 2

To solve for the inverse, let y = f(x). Then

y = 2^x

The inverse is solved by transposing the y and x variables and solving for y. Let's transpose:

x = 2^y

To solve for this, we have to take the log of both sides. Here, we're going to take the log[base 2] of both sides.

log[base2](x) = log[base2](2^y)

Using the log property that we can bring down a power outside of the log,

log[base2](x) = y*log[base2])(2)

Remember that log[base2](2) is calculated by asking yourself, "2 to the what power is equal to 2?" to which the answer is 1.

log[base2](x) = y[1]

Therefore

y = log[base2](x)

Answer 3

Sorry but I just cannot figure that one out.

Answer 4

f(x) = 2^x
x = 2^(f(x))
log(2)x = f(x)'

ANS : f(x)' = log(2)x